- Email: [email protected]
Two-dimensional radiative transfer in a finite scattering planar medium
TWO-DIMENSIONAL RADIATIVE TRANSFER IN A FINITE SCATTERING PLANAR MEDIUM? A. L. CROSBIEand J. W. KOEWINGS Thermal Radiative Transfer Group, Department of Mechanical and Aerospace Engineering,University of Missouri-Rolla,Rolla, MO 65401, U.S.A. (Received 29 September 1978) Abstract--The source function, radiative flux, and intensity at the boundaries are calculated for a twodimensional, scattering,finite medium subjectedto collimated radiation.The scatteringphasefunction is composedof a spike in the forward direction super-imposedon an isotropicbackground.Exact radiative transfer theory is used to formulate the problem and Ambarzumian’s method is used to obtain results. Using the principle of superposition,the resultsfor any step variation in incidentradiation are expressedin terms of universal functionsfor the semi-infinitestep case. Two-dimensionaleffects are most pronounced at large optical thicknessesand albedos.
INTRODUCTION
AN EXTENSIVE survey (‘) of multidimensional radiative transfer in a scattering medium reveals a lack of exact analyses and results. Recently, two-dimensional radiative transfer in a semiinfinite medium has been investigated. CROSBIEand LINSENBAIU-# studied a rectangular geometry, while CROSBIE and DOUGHERTY’* analyzed a cylindrical geometry. The source function, radiative flux, and intensity at the boundary were expressed in terms of a generalized H-function and were calculated for a wide range of parameters. Two-dimensional effects were most pronounced for conservative scattering. The objective of the present investigation is study of two-dimensional, multiple scattering in a finite medium exposed to collimated radiation. Specifically, the source function, radiative flux, and intensity at the boundaries will be expressed in terms of the generalized X- and Yfunctions. The primary objective is to obtain and present exact numerical results. These “bench mark” solutions should be extremely useful in evaluating various approximate procedures for handling multi-dimensional radiative transfer. PHYSICAL
MODEL
AND GOVERNING
EQUATIONS
The present investigation is concerned with two-dimensional radiative transfer in a finite, absorbing-scattering, planar medium. The medium and coordinate system employed are shown in Fig. 1. The following assumptions are used throughout this study: (1) steady-state intensity and temperature, (2) two-dimensional geometry, (3) no emission, (4) coherent scattering, (5) homogeneous medium, and (6) refractive index of unity. Although monochromatic radiation is considered in the analysis, the frequency subscript has been dropped to simplify the notation. The two-dimensional transport equation can be written as IP(cos @)dp’ d&.
(1).
The radiant intensity, Z, is a function of location (y, z) and direction (p = cosine of the polar angle t9and 4 = azimuthal angle). The extinction coefficient, &, is the sum of the scattering and absorption coefficients, a, + K. The scattering phase function is assumed to be a spike in the forward direction superimposed on an isotropic background, i.e. P(cos 8) =47r@(/.&- /.L’)S(f$- 4’) + I- f, tsupported, in part, by the National Science FoundationthroughGrant ENG 74-22107. SPresentaddress:WestinghouseElectric Corp., West Mifflin, Pennsylvania, U.S.A. 573
(2)
A. L. CROSBIEand J. W. KOEWING
Fig.
I. Definition
of the coordinate system used.
where 8 is the angle between the incident (P’, 4’) and scattered (IL,9) radiation; f is the fraction of radiation scattered in the forward direction; 6 is the Dirac delta function. The quantity f is also equal to the asymmetry factor. The utility of this phase function has been discussed by JOSEPH, WISCOMBE,and WEINMAN.O) The strong forward component is characteristic of the scattering phase function for large particles. Substitution of Eq. (2) into the transport equation and definition of the optical coordinates and albedo as
allows the transport equation to be written as cosfY$+sinBsin$-$+Z=S, .? Y
(4)
where the source function is defined as 2s
S(7yr
72)
=
-g
II 0
+1 I(T~,
T,,
P’,
4’)
dcl’
(5)
W.
-1
Equations (4) and (5) are identical to those for an isotropic scattering medium. For a non-absorbing (conservative) scattering the albedo o is unity, and for a non-scattering medium w is zero and ry = KY and rz = KZ. The medium is exposed to collimated radiation, i.e.
where ~0 is the cosine of the incident polar angle. Note the flux incident on the medium is IO&r,.). The other boundary is not exposed to radiation, i.e. Z-(rP 70,CL,4) = 0. Solution of the transport Eq. (4) and substitution into Eq. (5) yields the following integral equation for the source function:‘4’ ?S(
’ T:; Tyv
ToMTy
-
T:,
Tt -
7;)
dT: dT;,
(7)
where a0 = l/p0 and the kernel is EI(Q 7) = ’s I,- KoU@?j)
dt
(8)
and K. is the modified Bessel function of order zero. The z-component of the radiative flux is given by %(Tyyr
-.
Tz:
TO) =
---
hf(Ty)
eXp
(-c~OT~)/~O+
__
257
‘OS(
Ty,’
7:;
TO)&(Ty
^-
-
T;,
7~ -
7;)
dr: dr;,
(9)
575
Two-dimensionalradiative transfer in a finite scatteringplanar medium
where *
(10)
dy dx.
Ko(xy&%%
The diffuse intensifies leaving the medium in the r,, rz plane are given by
I-( ry, 0, u, ao) = u
z+tq, 70, u, uoo) = fJ I
(11)
S(rY,7:: ro) exp (-UT;) dr:
I0
TO
S(T~,7:: ~~)exp [-u.(T~-
4
(12)
dr:,
0
where u = 11~. In general, f(~~)
can be expressed mathematically with Fourier integral theory as (13)
where g(B) = &j:
(14)
f(rJ exp (-Z&) d7, (D
The source function, z-component of flux, and the intensities leaving the media can be written asg4) ozo -
qAq,
G:
TO) =
-1
ZO
Q,AT,,
UO;
TO) g(B)
ev
(iSTy)
(16)
dS,
I
z77yr
0,
a9 uo)
=
uzo G
m _-coWT
UO:
70) g(B)
exp
MW
WY
(17)
00:
TO) g(B)
exp
WY)
dS,
(18)
I
z+by,
70, WV 00)
=
uzo G
m T&T _ DI
I
JB(7*,
a:
ro)
=
exp (-UT=)+;
QB(rz,
0:
ro)
=
exp (-or&
I0 +T
TO J@(T:, a:
I0
(19)
TO)~,([G - T:),PI dd,
” J@(rL,u:
TO) sgn
(7,
-
WMh
-
T:[,B) dd,
(20)
9 &(a.
uo;
J8(&
TO) = o
uo: TV) exp
(-a;)
d&
(21)
I 0
T&r, uo: 70) = o
I0
” JB(& UO;ro)exp [-u(~~-
721 dr:.
(22)
The generalized exponential integrals are defined as %(r, B) = Imexp (-+r&!(dd@$%,
(23)
exp (-q/m)(dtlt*).
(24)
Physically, the quantities, (oZo/47r>.Ze exp (i/3rY), ZOOSexp Ci/3T,>,(uZ0/47r)RBexp (iPry), and (uZd47r)Tg exp (i/?Ty) are the source function, the radiative flux, the reflected intensity, and the transmitted intensity, respectively, for a medium exposed to cosine-varying collimated radia-
A. L. CROSBIE and J. W. KOEWING
576
tion, i.e. Z&L - pO)S(d) exp (@TV). Inspection of Eqs. (15)-(22) reveals the importance of the function _Z@(T~, a: rO).In the next section, Ambarzumian’s method is used to solve Eq. (19). AMBARZUMIANS
METHOD
Source function
In order to develop the integro-differential equations for the cosine-varying, collimated source function Je at the boundaries of a finite medium, a series of transformations is made. This procedure parallels that of CROSBIEand BREIG.(‘*‘) Let 7z be replaced by 7. - 7z in Eq. (19) to give J@(T~- T,, CT:TV)=
e-"("-~)+
!f 2
I
b"
J8(n-
T:, CT:n) kf,(lr, - 73, B) dr:.
(25)
The key step in the procedure is to treat the optical thickness as a variable. Differentiation of Eqs. (19) and (25) with respect to r. yields ~(~z,~;~O)=~J
( 2
fi
aTO ~(To-w;~o)= aTo
To,
a; 70)%(To-
725PI + y
_ IaaJ,(Tivu;T”~‘&(Tz
T;(,
o
- u eedzo-Q) + T J,(O, a;
TO)
‘0S( +o 2 Io aTo
‘&(IG- 4, P) W
7:, u ; To)
To -
~I(TO-
fj)
dT;.
aTo T,,
(26)
/3)
(27)
At this point in the development, a function QB(rz: ro) is introduced which is defined by
(28) Replacement of
TV
Q(TO
by 7. -
TV
- ~2: TO) =
and 7: by
7. -
T:
; ~%'I(To - ~z,/3)+ ;
gives
I” @s(~o- 7:: ~0)%‘,((T,- ~1,/3) d7:. 0
(29)
Multiplication of Eq. (29) by Jg(~o, u: ro) and comparison of the resulting expression with Eq. (26) yields $@(“’ u; “) = J@(T~.a: ro) @B(~o- TV:TV).
(30)
0
A similar approach applied to Eq. (27) produces
&(70 -
Tz, u:
To)
=- uJ@(~o- rrr a: TO)+ Je(O, a: TO)(PB(~o- ~z: uo).
aTo
(31)
Replacement of u by v(t* + j12) in Eq. (25), multiplication by (o/2) dt/q( t2 + p2), and integration from 1 to 03yields a$(70 - rz: To)=
02
JP(TO Im I
7,
VU2+ ~‘1:~0)$:
B2).
(32)
Setting TV = 0 in Eqs. (30) and (31) and using Eq. (32) produces the following equations for the collimated source function at the boundaries: 2”’
u’
To) = J,(T~,
a:
TO) Q8(To:
To),
0 S(To,
aTo
u;
To)
=
-uJ@(n,, a: TO)+ Ja(O, CT:TO)G$(To: TO).
(33) (34)
511
Two-dimensionalradiative transfer in a finite scatteringplanar medium
Replacement of CLby d/(1 +/3*)/g and x by q( 1 + fi*)/~(t* + fl*) in these equations and the introduction of the generalized X, and Ys functions, X&L; TV)= &(O, g(l + p*)/p: $ and Yh:
70) =
J,~To.
~41
+P*)/P;To),
yields
!z!+; To) = Y&&h: TO)qg(70: To), 370 aY,(cc;To)_ -x41 + p*) 370
-
CL
(35)
Y& : To)+ X&L To)@‘8(70: To),
(36)
where Q/3(70;70)= ; V/(1 + p*) 1’ 0
Y,(x: TO)dx
(37)
xV(l + P2(1 - x2))’
The X, and Ys functions are the two-dimensional analogs of the famous X- and Y-functions of Chandrasekhar. With initial conditions J,(O, cr; 0) = J,(T~, a; 0) = X,&L; 0) = Y&L; 0) = 1, Eqs. (33)-(37) are the basic equations used for evaluating the cosine-varying collimated source functions at the boundaries.
Reflection and transmission functions
In order to relate the reflection and transmission functions to the source functions at the boundaries, a series of transformations must be made. Differentiation of the integral equation for the source function (19) with respect to TVyields
g&T,, 6 70) aTz
= -ueemz +3 Ja(O,a: ro) WT,, p) -t
” !%.I (‘ivu:‘O)$,(lTz +o 2 I0 ar:
_ T;l,
p)
Jg(~o,U: TV)$,(T~ - T,, p) dT;.
(38)
Comparison with Eqs. (28) and (29) gives
aTz
= -u&(Tz,
U:
70)+ J#(o,0:
To)
@‘8(7,:TO)- J,(To,U: To)@@(To-Tz:
To).
(39)
Substitution of u = a0 and multiplication by exp (-orz) and integration form 0 to 7. yields (a + uo)R&~go;To)= Jp(o,co; To)
To @,q(T,;To)e--
dT,
TO @'a(~0- TZ:ro) e--
dT,
1
(40)
with the help of the relation (which is obtained with integration by parts) =
e-c,.ro
&(To,uo:70)- J&h co: To)
+
uR&,
UC,:ro).
(4
Evaluating the equation for the source function (19) at the upper boundary (T*= 0) and interchanging the order of integration yields J,q(T:,
U: To) e-T;V(‘*+B2)dT;$l;
I
@*).
(42)
The inner integral is the reflection function Rp The symmetry of Rp with respect to u and was shown by BREW and CROSBIE.(7) Utilization of the symmetry of RB, interchange
UO
Two-dimensionalradiativetransferin a finite scatteringplanarmedium d( 1 +
579
/3*)/~(t* + p*) leads to
u@o,a:
I
I
xMxY B, 2 0 x+~(1+B2)/a
To)= 1 - 0
- YB(x: ~0) Ja(O,u:
[Xs(x: 70)J,(O, a: 70)- YB(x: TO)&(To, a: TO)]dx,
TO)] dx,
(54)
(55)
where $,(x, p) = (1 + 8*)/U + /3*(1- x2)P2.
(56)
DISCUSSION OF NUMERICAL RESULTS
Cosine-varying
results
The integrodifferential equations (33) and (34) were used to calculate the source functions at the boundaries [&(O, a: TV) and J@(T~, a: TO)], while Eqs. (54) and (55) were used to calculate the fluxes, Qs(O,a: TV) and C&(T~, a: TV), at the boundaries. The numerical procedure is outlined elsewhere.@’ Numerical results are presented in Tables 1 and 2 for a wide range of parameters. A more extensive tabulation is available. ~3)in the discussion that follows, the figures and discussion pertain to normal incidence, u = 1.0. The influences of optical thickness TV,spatial frequency p, and single scattering afbedo w on the source function are illustrated in Figs. 2-4. The source functions for two different albedos exhibit the same general trends; however, the source function for an absorbing medium (o < 1) is less than that for the purely scattering medium (w = 1.0). As might be expected, the influence of albedo is most pronounced at large optical thicknesses and depths and at small spatial frequencies. Except for very large optical thicknesses and albedos near unity, the source function has converged to the one-dimensional result (/3 = 0 for B < 0.01. For 7.5 10.0 and o S 1.0, the maximum percentage error at p = 0.01 is 0.03 for T = 0 and 0.2 for T = TV. As the spatial frequency becomes large, the source function becomes independent of albedo and approaches exp(-or). This result can easily be seen from Eq. (19), since &(T, p) goes to zero as /3+ w At /3 = 10.0, 7. = 10.0 and o = 1.0, the percentage error between the source function and exp(-or) is 0.23 for T = 0 and 4.0 for T = TV.
Fig. 2. The source functionfor the cosine-varyingcase.
Two-dimensional radiative transfer in a finite scattering planar medium
Table 1 (Confd) w=l B
0.3
0.001
T
0
= 0.5
J&bx~,)
J.&,,u;T~)
1.496624 1.496824
1.053783
u
=
1
Qp.a;To)
zx:
01797494 0.797493 0.797492 0.797491 0.797480
0.004 0.008
:%x23 1:053782 1.053775
E% 0:030 0.040
:%%35 11053667
0.797524
W8 ghg
:'m~: I:053320 1.052960
"o%s's:~ 01797614 0.797707
0.002
ym;r;
0.797152
1:4A55R9 1.4774R6 1.467776
sz::t 0:792343
g:
Q,(~J;T,)
xsR:67:gf 01779900 0.772530 0.763121
i&8 0.965742 0*935500
:+gx 1:soo 1.750 2.000
iK25'x 01736994
"o-X%~ 0:859463 0.821572 0.791716 0.767965 0.748130
3-z% 3:500 4.006 4.500 5.000
8=‘73%1 0:700420
t:f%
0.916104 8'XX~os 0:942991 0.940677 0.640408 0.633421 0.620507
:%m(: 1:039071 1.011290 1.026093 1.012375
:%m 1:003913 1.003129 l.CO2606 1.001954
1.000390 1.000260 :':oooo:s~ 1:rooo97 l.OI?OO78
0.631626 0.626521 00%::: 0:618798
0.900545 O.QA3715
~%5550' 0:619311 0.617088
Km 0:991764
8*x533: 0:611695 0.6103R3
KG;260 0:607291
0.607720 0.607481 0.607321 0.607124 0.607005 8'%Kg7 0:606689
8%W 0:606910 0.999667 0.999750 Kz%z 0:999900 "0*:;;9'3; .
0.606561
581
582
A.L.CROSBIE andJ.W. KOEWING
TableI (Contd) w=l
B
J&‘hw,)
T
0
=
1.0
Jg(To.U;~o)
a=1
Q&bJ;To) 0.658671 0.658671 0.658672 0.658673
:‘ws I:757372 1.757368 1.757351 1.757339
Oo%x .
Qg(To.O;~o) 0.658671 0O*txs: 0:658669 0.658663 0.658658 0.658620 0.650556 0.650467
0.659916 0.663557 8%159e)6322 a:665717 :%%5;; 1:546481 1.485017 1.432099 1.387346
0.647817
Ptf25:$ 0:737028 0.554517 0.532950 0.514346 0.864362 0.880189
8't%:i; 0:454709 0.440849 ym;q0
km% 0:404710 0.400556
0.943340 0.94R323
H8%38 01395079 0.391600 0.388903 0.304242
0.377962 0.376420 0.375120
0.980545 0.983715 0.985996 0.907717
0:4156?0 :%%32 1:077707
1.026093 1.022375 1.019584 :-g#i; 1:007e33 1.006265 1.005219 :%%:z 1:003129 1.002606 1.001954 1.001563 1.001302 l.PFO977 1.000781 1.000521 l.COO390
0.369537 0.369328 0.369036
0.997154
0.369366 0.369116 0.36R938 0.3bRROS 8*3%% 0:368341
x%x! 0:999500 0.999667
:%x85 1:000155 :%% .
x%58; 0:377699 0.375631 0.374203
0.999950
00'33%$! 0:369110 0.368064 ix%: 0:367941
Two-dimensional
radiative transfer in a finite scattering planar medium
Table 1 (Contd)
3:801 0.002 :%z 0:oto 0.020 0.030 x40” 0:060 8%8 0:200 0.300 0.400 0.500 0.600 I)%: 1:zso
;454907 :Wo963 _449AS4 ;44se44 .443204 -434058 .422571
0.449073 0.446783
SW%
0: 28704 0.216200 0.159183
:Wdl! .761252 .648909 .545135 :tPW .364126 .296972
3.500 4.000 4.500 5.000 6.000 7.000 8.000 9.000
25.000 30.000 g.38; 60:000 80..000 :x% I so:ooo 175.000 200.000 250.000 300.000
:1:X .190361 .I69914 .I53412 .12R410 :WZ .006227 .077?07 : 8%;:; .044620 .039072 :83K5 .077375 :::%!69: .009795 .007833 :cdoO%~~ .004473 .003913 .003129 : oO%%t .001563 :%g’% .000781 -009521 .000390 :8X% %8&G .000070
0.761456 0.261989 0.262873 8.3% 80’ 0: 267590
oo%% 0:434310 00. y&5 k
8”4m a:540311 0.614646 0.669164 0.718963 0.755573
“6W~f
0:039039 0.024341 0.017696 XlX 1:0514E-02 9.5407E-03 8.9626E-03 8.5800E-03 ;. 3;u&-cg
xm’: 0: 838381
7.-I 279FIO3 7.6425E-03 7.5109F-03
0.924889 0.934764 0.942332
7.4126E-03 7.3365E7.2047F-03 7.1204E-03 7.0619E-03 7.0190E-03 6.9600E-03 6.9215E-03 6.8944E-03 6.R742E-03 6.0779E-03 6.0050E-03 6.7914E-03 6.7806E-03 6.7734E-03 6.7601F-03 6.7645E-03 6.759lE-03 6,7556F6.7512E-03 6.7465E-03 6.7467E-03 6.7445F-03 6.7432F-03 6.7415E-03 6.7406E-03 6.7397E-03 6.7393E-03 6.7390E-03 6.7386F-03 6,7385E-03
03
8*W86” 0:9a77 0.991764 “0*9’83!$Z 0:496Q23 0.9966132 pmg 03
0: 990006 8%KI 0:999001 0.999168 0.999376 on-99Z28V 0:999750 0.999933 0.999875 “o-Z~o3~ 0:999950
17
583
A. L. CROSBIEand
584
J.W. KOEWINC
TableI (Contd) w=l 8
T
0
=
10.0
J6(0.wo)
J&T~AJ;T~)
Qp,o;To)
Q&~o’wTo)
Km3
0.254591
FP ::tz 0:147013
0.146995 0.146992 0.1469'83 0.146948
2: 653174
0.147O66 0.147277
2.651079 2.652695
7.652400 2.650022 2.64bC70 7.640620
I%W’8~ 2: 605303
:*x:i
1:460519 1.410037 1.364126 1.296972 1.250419
%o"o"
6:OOO
7.000 8.000 9.000
:*"0%320~ 1:062309
1.052002
25.OOO 30.000 35.000 40.000
60.000 00.000 E%% 150:000
175.000
:%G%8 l:O31290 3%9833 l:O19504 1.013062 :%m 1:006265 1.('05219 1.004473
5~00-0000s :*:"03;:: 3oo:ooo 1:OO2606 %x008
bOO:OOO 000.000
:‘K%m 1:001302
1.000977 :%%51 l:ooO39o pgx.P~~ 1:000155 :%800tX .
8.
0.
mm;
lt 45808
i? :Et% 0:139799
8%% 0: 125655
"o.mm;;
2.000 2.500 3.000 3.500 4.000
0=1
1:3555E-02 b.Z163E-03 1.4604E-03 4.6981E-04 1.9465E-04 1.2434E-04 9.7615E-05 0.4170E-OS 7.0052F-05 b.4205F-05 b.O39OF-05 5.7017E-05 5.5993E-05 5.4633E-05
5.2744F-05 5.1495E-05 5.0608E-05 4.9946E4.9433E-05 4.8545E-05 4.7977E-05 4.7583E-05 4.7293E-05 yg9g-CJ;
0.173361 0.106750
w$35$ 0:121039
8%~~~: a:429574
0”:83%8 0.017844 0.3024E-03 3.0990E-03 9.7414E-04 3.3173F-O4 1.4744E-04 9.9098E-05
8*f%P8 O:bl4365 0.669109
A.O636F-05
7.1305E-05
%Q% 0:892OSl 0.902721 0.911450
05
4164 2 4Ez05 4.6310E-05
4.6006E-05 4.5052E-05 4.57bOF-05 4.5687G05 4.5639E-05 4.5604E-05 t. ym;-p
4:5519ErO z 4,5489E-05 4.5471F-05 4.5459E-05 4.5444E-05 4.5435E-05 4.5424E-05 4.54 1 BE-05 4.541 ZE-05 4.54096-05 4.5407E-OS 4.5404E-05 4.54035-05
0.962080
0.968133 0.972518 0.975841 0.900545 8%I53X 0:907717 0.991764 0.993805 cm%
0:996682
8%X8 0.999376 0.999500 0.999667
0.999950
b.?391F-05 5.7@49E-05 5.5233F-05 5.3515F-05 5.2303F-05 5. ICOQE-05 5.0159E-05 4.9339E-05 4.0760E-05 4.8328E-05 4.7995E-05 4.7419F-05 4.7052E-05 4.6798E-05 4.66 12E-05 4.6357E-05 4,6190E-05 4.60736-05 4.598bE-05 4.57866-05 4.5688E-05 4.5630E-05 4.5503E-05 *.5553l=-05 4.5531F-05 4.5514E-05 4.5491E-OS 4.547bE-OS 4.5457E-05 4.5445E-05 4.543nF-os 4.5420E-OS 4.5423E-05 4.5415E-05 4.5411E-05 4.5407E-05 4.5406F-05 4.5404E-05 4.5403E-05 4.54026-05
585
Two-dimensionalradiative transfer in a finite scatteringplanar medium
Table 2. Sourcefunction and radiative flux at the boundariesfor the cosine varying case with rO =
I and
(T= I.
w-o B
.99
To
=
1
u = 1
JB(~oq,)
gpdJ;T,)
0.950115 0.950115 0.950114
0.667070 0.667078
WW 0: 950082 00. y9e95 a:94959
3
0.080
8%W8 9:667377 0.667509 0.667842 0.668269
Q&‘aqJ 0.650977 i?6”5oo~T77 0:650976
0.650673 0.650539 0.650199
0.649764
0.600 P-888 I:250
:- WEi 0:765826 0.786689 0.804923
t-%8 2:ooo kZ88 3.500 4.000
4.500 5.000
12.500 Wo"oO
80.000 100.000
125.000
o0wwi 0:980749 0.983884
0.377797 0.37 332 0.37 s 243
0.986141 0.987843
wt3z 0: 370769
00-m% 0: 995086
8:338 #8t 0.3695 0 0.3693 T4 0.369025 “omx 0: 360450
0.999938 0.999950
QSRT Vol. 21, No. 6-G
0.372580 8*33839"2" 0: 369724 0.369351 0.369104
A.L.CROSBIE~II~J. W.KOEWING
Table2 (Cod) w =
Jpd
0.90
PO)
1.597 04 m 8; 1:597 f00 1.597 89 1.597 180 :.g'dg 13596822 1.596606
To
=
1
JB(~Os~:~O)
u
=
1
Qs(O,o;TO)
0.827278 0.827278 0.827277 8.e8tzEE 0:8272SS 8*8e$3& 0:826912 0.826706
Elm l:S9482S
kV358tJ
:.xE l:S62393 1.545427 1.526728
0.7so491 0.757087 0.771386 0.786014
1.326233 ;.m$*~;
0.566512 O.SS1877 8'E8E? o:so315s 0.489033
1:213 !98 ytg;s; 1:1484 z 1 1.134546 '1.most 1:085836
0.420011 0.411401
?8Mf 1:055S41 1.04t429 1.C39884 1.03495s
0.966229 0.971570 yg*g o":31"8%i
:.0at?M 1:020069
1.004694
:.E%f 1:000468
0.922633 0.933961
0".371%% 0:374556
0:982 3 81 0.985406 0.987443 0.988981
8.m: 0:37oso*
fmosi 0:99S538
8.m: .
Oo.%%P% 0:997441
0.368312 0.368203
0.999251 0.999438 iEGm8 0:99977s Oo.z%tB 0:999910 0.999944 0.99995s
00'377%P 0: 76657 3 8.3%% 0:37277C "0*338% .
Two-dimensional
radiative transfer in a finite scattering planar medium
587
Table 2 (Contd) w =
8
J6(0,u;~,)
0.50
T
0
=
J&T~,u;T~)
1
cl=1 Q~hJ;70)
i%OL :‘St:t%
8-885 0:ooe
tii446058
0.446058 0.446058
1:22bLbA
wx25 oo%$ 1:226144 0:030 1.226114
0.040 0.050
0.527492 0.527475 0.527447 0.5374OR
oo%% 0: LOO 0. ZOO
1.000
:-W86’ 1:220958 1.217200 y;l2”6;;
0.515091
1: 196466 1.184800
:%8
1:750 0.433440
6.000
1.058229
T*K% 91000
:‘85ocx 1:040389
“o%z’; 0: 387301 8. g5pfJ;
koO/%S
0:37
::82’:88:
8: GX4fZ 0.375318 0.373795
0.900882
-0.900988 0.901048 0.901544 0.902345 0.9034 00-:8x$ 0:909b 0.913312
18 11
Oo-4?bZ 0:436766 8-WE4 0:423135
“o-X% 0:927LLL 0.931228
:-:x2 0:407565
0.377683 0.375624 p;;w;‘:
3 991
0.9905
1.006467 ::MX 1.003118 1.002599 1.002229
Q&To'UX,)
0.446058
1.226169
19
01372628 8% t “o:t 0:370556 0.370216 “0-366x: 0:368805
0.368844
0.998348
:%.%z 1:001301 1.000976
8*XK 0: 368059 “0*3x5;
0:367951
“0-m:; 0: 367908 0.367897 0.3b7894
:-
3tzo5
033b7902
8-3X8 0:367089
588
A. L. CROSIIIEand J. W. KOEWINC
B Fig. 3.
Fig. 4.
Fig. 3. The influence of albedo on the source function for the cosine-varying case and TO= 1.0, u = 1.0. Fig. 4. The influence of albedo on the source function for the cosine-varying case and 7s = 10.0, u = 1.0.
Under certain conditions, the source function for a finite medium can be approximated by the source function for a semi-infinite medium. The maximum difference between the finite thick and semi-infinite source functions occurs for the one-dimensional case (/I = 0) and for pure scattering (w = 1.00). As the albedo is decreased or the spatial frequency is increased, the difference is drastically reduced. For /3 = 0 and 7. = 10.0, the difference is 0.25 at both boundaries for w = 1.0 and is 0.000022 at the upper boundary (r = 0) and 0.0093 at the lower boundary (T = ro) for o = 0.90. In general, the difference is always smallest at the upper boundary. For the upper boundary, the largest difference is 2.2(IO)-’ for o 5 0.9 and 7. 2 10 and 1.3(10))’ for 7. 2 0.1 and /3~~ z 10.0. The influence of optical thickness, spatial frequency, and single scattering albedo on the radiative flux at the boundaries is illustrated in Fig. 5. Like the source function, the radiative
Fig. 5. The radiative flux for the cosine-varying case.
Two-dimensionalradiative transfer in a finite scatteringplanar medium
589
fluxes leaving the medium decrease as the spatial frequency increases. The radiative flux leaving the upper boundary is (l/a) - (IJO, a; TO). This behavior is simply explained by the fact that the amount of radiant energy per unit time, per unit length incident on the medium in a strip (-Q, 5 TV 5 rb) is 2 sin (&,)/(u/3) and decreases with /3. As would be expected, the flux leaving an absorbing medium (w c 1) is less than that leaving a purely scattering medium (o = 1). When the spatial frequency becomes large, the radiative flux becomes independent of albedo and approaches exp (-UT)/U. An increase in the optical thickness increases the flux leaving the upper boundary and decreases the flux leaving the lower boundary. Semi-infinite step results The numerical results for the semi-infinite step boundary condition, i.e. I
(57)
= i sgn (T& and g(B) = -i/(2,$),
are presented in Tables 3-5. These results were calculated from the following equations:
R(T~, a, uo: TO) =
;[R,d
-
a, uo: To)= -i
T(Tyr
a, uo: TV) sin(B~~)y,
omT&6 00: TO)sin (@TV) $,
(5gd)
S(7y, 7Z;To)= (010/47r)J(Ty, 72,uo: To),
(59a)
I
where
4*(Tyr72: To) I-(~yr
=
0, a, a01 =
ZOQ(Tp
Tzr uo:
To),
@W
Wo/47r) R(~yra, UO;7oh
z+(TyrTara, uo) = (UZol4~)T(Ty,
a,
co;
(59c)
To).
094
The numerical procedure is outlined by KOEWING. (‘) All the results are for normal incidence, uo= 1. This boundary condition is important because other boundary conditions can be expressed in terms of it. The spatial variation of incident radiation can be approximated by a series of steps, i.e. f(ry) = 6 for ?k < ry < Tk+l with k = 1,2,. . . M, where fk is a constant between zero and unity and where 71= --03and T M+l = +Q). This boundary condition can be written as
and the function g(B) becomes
g(B)= (d2@)f =Ifk[-exp (-i/.bk) + exp (-i@Tk+l)].
(61)
Thus, the source function, flux and intensity become S(ryr 7~:70)= (m&,,4a) M fk[J(T,, F=I q*(q,
7r: 70) =
lo
M fk[Q(Tyc =I
Tkr 72, uO:TO)
Tk. Tz, UO: TO)-
Q(Ty
--J(Ty
-
-
Tk+lr Tz,UOo; 70)],
Tk+l, Tz, 00:
To)],
(624
(62b)
A. L. CROSBIEand J. W. KOE~V~NG
590
Table 3.Sourcefunctionandradiativefluxatthe boundariesforthe semi-infmitestepcase witho = and o= 1. T
Y
J(T~.O,~;T~)
J(T~,~,&;T~) T
0.0
0.50000
0.01 0.02
0.50968 0.51591 0.52856 0.54107 0.55427 0.56809 0.57572 0.57860 0.57936
0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
0.57937 0.57937
0
=
Q(T~.O.~;T~)
Q(T~,T~>~;T~)
0.1
0.452423 0.46160~ 0.467623 0.479983 0.49237E 0.50551E 0.520123 0.526943 0.529903 0.53058E 0.53060E 0.530603
00 oo 00 00 00 00 00 00 00 00 00 00
0.50000 0.49482 0.49182 0.48648 0.48234 0.47922 0.47706 0.47641 0.47620 0.47616 0.47616 0.47616
0.452423 0.457373 0.46032E 0.46565E 0.469853 0.47303E 0.475243 0.475903 0.476113 0.476163 0.476163 0.476163
00 DO 00 00 00 00 00 00 00 00 00 00
0.50000 0.49260 0.48739 0.47565 0.46189 0.44408 0.41942 0.40627 0.40024 0.39878 0.39875 0.39875
0.303273 0.30874E 0.312863 0.32268E 0.33499E 0.35203E 0.37720E 0.390973 0.397223 0.39872E 0.39875E 0.39875E
00 00 00 00 00 00 00 00 00 00 00 00
0.50000 0.49188 0.48594 0.47203 0.45469 0.42994 0.30775 0.35661 0.33658 0.32955 0.32934 0.32934 0.32934
0.18394E 00 0.18820E 00 0.19165E 00 0.200433 00 0.212573 00 0.231943 00 0.26996E 00 0.30120E 00 0.321923 00 0.32912E 00 0.329333 00 0.329343 00 0.329343 00
0.50000 0.49107 0.48433 0.46799 0.44662 0.41386 0.34844 0.28305 0.21359 0.14767 0.13209 0.13065 0.13064 0.13064
0.336903-02 0.38178E-02 0.425153-02 0.55242&02 0.75983E-02 0.11648C01 0.232703-01 0.410573-01 0.703073-01 0.114863 00 0.129203 00 0.130633 oc 0.130643 00 0.130643 00
0.50000 O-49099 0.48417 0.46761 0.44585 0.41233 0.34461 0.27543 0.19863 0.11432 0.08237 0.07404 0.07349 0.07350
0.22700E-04 0.131093-03 0.239373-03 0.56402E-03 0.11047E-02 0.218513-02 O.S4149E-02 0.107343-01 0.209363-01 0.455033-01 0.654373-01 0.729583-01 0.734993-01 0.734993-01
T = 0.5 0
0.0 0.01 0.02 0.05
0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
0.50000 0.51306 0.52266 0.54522 0.57327 0.61270 0.67636 0.71843 0.74170 0.74827 0.74841
0.74841
0.303273 00 0.312683 00 0.319973 00 0.337883 00 0.361353 00 0.39618E 00 0.45597E 00 0.49710E 00 0.52020~ 00 0.52675~ oo 0.52689E 00 0.52689E 00 T = 1.0 0
0.0 0.01
0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 20.00 100.00
0.50000 0.51419 0.52492 0.55089 0.58456 0.63498 0.72790 0.80373 0.85746 o.87801 0.87868 0.07869 0.87869
0.183943 0.19111E 0.19700E 0.212243 0.233723 0.269023 0.34290E 0.410693 0.462073 0.482353 0.483033 0.483032 0.48303E
00 00 00 00 00 00 00 00 00 00 00 00 00
T = 5.0 Cl 0.0 0.01 0.02 0.05 0.10 0.20
0.50 1.00 2.00 5.00 10.00 2c.00 50.00 100.00
0.50000 0.51551 0.52757 0.55751 0.59779 0.66137 0.79274 0.92652 1.06730 1.19578 1.22494 1.22760 1.22762 1.22762
0.3369OE-02 0.41175~-02 0.484253-02 0.697293-02 O.l0450E-01 0.17252E-01 0.36847E-01 0.67054E-01 0.11741E 00 0.19635E 00 0.222543 00 0.225173 00 0.225203 00 0.22520E 00 = 10.0 TcJ
0.0
0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 20.00 50.00 100.00
0.50000 0.51565 0.52783 0.55816 0.59910 0.66401 0.79931
0.93960
1.09301 1.25322 1.31088 1.32565 1.32661 1.32661
0.22700E-04 0.208323-03 0.393773-03 0.94983E-03 0.18760~-02 0.372663-02 0.926053-02 O.l8379E-01 0.35892E-01 O.V3309E-01 0.11310E 00 0.12634E 00 0.127303 00 o.12730E 00
I
Two-dimensional
Z3Ty, 0, u9 go) = Wo/49r) M fdR(q c =I
z+(ryr70, uv u0) =
591
radiative transfer in a finite scattering planar medium
(dO/4r)
- Tk,a, uo; 70)- R(q - n+1r a, go;
kz,fk[T(Ty - rk,u:uo:
TO)- T(Ty - 7k+lr&uO:
7011, TO)].
In using the above relations, the following identities are useful: J(-T~, T,, a: 70)= --J(T~,T,, a; 70)and Q(-T~, T,,u: R(-T~. u, uo; TO) = -R(T~,u,
UO;TO)
and
TV) = -Q(T?, T,,U: TV).
T(-T~,u,u~:T~)=
J@s T,, CC70)= 5 J~=o(T,,u: TO)and Q(Q),T,,a:
-T(T~,u,
ro) = i C&,(T~,
NC%u, UO:TO)= i R,&u, uo; TO)and T(m, a, ~00:TV)= i
u~;T~),
U; ro),
Tfico(u,uo: ro).
By evaluating the functions J, Q, R, and T, a wide range of problems can be solved.
Table 4. Source function and radiative flux at the boundaries for the semi-infinite step case with r0 = 1.0 and o= I.
w = 0.99 0.0 0.01 0.02
0.05 0.10 0.20
0.50 1.00 2.00
5.00 10.00 100.00
0.50000 0.51400
0.18394 0.19100
0.52457 0.55012 0.58321 0.63267 0.72353 0.79732 0.84927 0.86897 0.86961 0.86961
0.19678 0.21174 0.23280 0.26735 0.33944 0.40533 0.45499 0.47442 0.47505 0.47506
0.50000 0.49199 o. 48614 0.47244 0.45540 0.43114 0.38994
0.35971 0.34043 0.33374 0.33354 0.33354
0.18394 0.18814 0.19152 0.20015 0.21205 0.23102 0.26813 0.29846 0.31842 0.32528 0.32549 0.32549
w = 0.90
0.0 0.01 0.02 0.05 0.10 0.20
0.50 1.00 2.00 5.00 10.00 100.00
0.50000 0.51234 0.52156 0.54366 0.57193 0.61353 0.68787 0.74583 0.78457 0.79821 0.79860 0.79860
0.18394 0.19004 0.19498 0.20765 0.22527 0.25377 0.31182 0.36303 0..39983 0.41325 0.41364 0.41364
0.50000 0.49292 0.48781 0.47596 0.46142 0.44110 0.40776 0.38457 0.37077 0.36641 0.36630 0.36629
0.18394 0.18759 0.19049 0.19783 0.20783 0.22352 0.25341 0.27677 0.29119 0.29570 0.29581 0.29581
0.50000 0.49643 0.49394 0.48839 0.48191 0.47347 0.46128 0.45435 0.45117 0.45045 0.45044 0.45044
0.18394 0.18568 0.18701 0.19026 0.19449 0.20076 0.21155 0.21871 0.22223 0.22302 0.22303 0.22303
w = 0.50
0.0 0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
0.50000 0.50619 0.51066 0.52102 0.53369 0.55131 0.57982 0.59910 0.60995 0.61302 0.61308 0.61308
0.18394 0.18682 0.18906 0.19459 0.20196 0.21326 0.23430 0.25066 0.26071 0.26368 0.26375 0.26375
@aI (6W
(63~) (63d)
A.L.CROSBIE andJ. W. KOEWING
592
Table 5.Reflection andtransmission function forthesemi-infinite step casewithq,= I andv= I.
T., 0.0 0.01
0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
T 0 = 0.20 R T
0.07520 0.07804 0.07987 0.08363 0.08754 0.09195 0.09729 0.09996 0.10118 0.10147 0.10147 0.10147
0.07421 0.07702 0.07884 0.08259 0.08648 0.09089 0.09622 0.09889 0.1001; 0.10040 0.10041 0.10041
.r = 0.50 cl
T0 = 1.00
T = 2.00 0
R
R
T
R
T
0.15837 0.16717 0.17379 0.18968 0.21012 0.24035 0.29495 0.33844 0.36870 0.38014 0.38051 0.38051
0.11627 0.12330 0.12872 0.14204 0.15964 0.18653 0.23726 0.27927 0.30914 0.32054 0.32091 0.32091
0.16625 0.17648 0.18440 0.20401 0.23026 0.27134 0.35414 0.43340 0.50550 0.54836 0.55153 0.55157
0.05851 0.06328 0.06723 0.07758 0.09247 0.11775 0.17548 0.23895 0.30353 0.34491 0.34807 0.34811
0.16705 0.17356 0.18919 0.20925 0.23888 0.29218 0.33441 0.36361 0.37454 0.37490 0.37490
0.12320 0.12853 0.14161 0.15888 0.18521 0.23470 0.27549 0.30430 0.31519 0.31555 0.31555
0.17630 0.18407 0.20326 0.22888 0.26885 0.34896 0.42496 0.49328 0.53305 0.53588 0.53591
0.06317 0.06702 0.07709 0.09153 0.11598 0.17151 0.23212 0.29314 0.33147 0.33429 0.33432
T
o .12g48
0.11933
0.13568 0.14010 0.15020 0.16227 0.17850 0.20322 0.21883 0.22729 0.22966 0.22971 0.22971
0.12518 0.12939 0.13908 0.15078 0.16666 0.19112 0.20668 0.21513 0.21749 0.21754 0.21754 w = 0.99
0.01
0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
0.07801 0.07981 0.08354 0.08739 0.09175 o.og700 0.09963 0.10083 0.10111 0.10112 0.10112
0.07699 0.07878 0.08249 0.08633 0.09068 0.09593 0.09857 0.09976 0.10005 0.10005 0.10005
0.13560 0.13997 0.14992 0.16182 0.17778 0.20204 0.21732 0.22557 0.22787 0.22792 0.22792
0.12511 0.12526 0.13882 0.15034 0.16596 0.18997
0.20519 0.21343 0.21572 0.21577 0.21577 w = 0.90
0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
0.07773 0.07935 0.08267 0.08609 0.08992 0.09450 0.09677 0.09780 0.09804 0.09804 0.09804
0.07671 0.07832 0.08163 0.08504 0.08887 0.09344 0.09571 0.09673 0.09698 0.09698 0.09698
0.13494
0.13880 0.14755 0.15790 0.17162 0.19206 0.20459 0.21117 0.21295 0.21299 0.21299
0.12448 0.12815 0.13654 0.14656 O.i5998 0.18019 0.19267 0.19924 0.20102 0.20106 0.20106
0.16599 0.12232 0.17487 0.06235 0.17165 0.18508 0.20208 0.22674 0.26975 0.30234 0.32367 0.33107 0.33128 0.33128
0.12692 0.13809 0.15263 0.17441 0.21414 0.24551 0.26651 0.27388 0.27409 0.27409
0.18143 0.19737 0.21823 0.24992 0.31053 0.36412 0.40785 0.42946 0.43060 0.43061
0.06546 0.07344 0.08464 0.10312 0.14331 0.18455 0.22264 0.24320 0.24433 0.24434
0.16214 0.16483 0.17098 0.17836 0.18838 0.20399 0.21406 0.21954 0.22105 0.22108 0.22108
0.11922 0.12136 0.12636 0.13254 0.14120 0.15535 0.16492 0.17027 0.17177 0.17180 0.17180
0.17035 0.17332 0.18018 0.18859 0.20035 0.21973 0.23350 0.24196 0.24468 0.24476 0.24476
0.06018 0.06144 0.06451 0.06852 0.07456 0.08591 0.09541 0.10218 0.10463 0.10470 0.10470
w = 0.50 0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
0.07655 0.07740 0.07911 0.08082 0.08269 0.08482 0.08593 0.08626 0.08637 0.08637 0.08637
0.07554 0.07639 0.07809 0.07980 0.08166 0.08379 0.08480 0.08524 0.08534 0.08534 0.08534
0.13230 0.13423 0.13849 0.14333 0.14945 0.15786 0.16251 0.16472 0.16526 0.16527 0.16527
0.12197 0.12381 0.12788 0.13256 0.13852 0.14682 0.15145 0.15366 0.15420 0.15421 0.15421
For example, the case of a strip of collimated radiation, i.e.
can be expressed in terms of the present investigation as follows: M = 3, f, = fJ = 0, f2 = 1, 71=-m, 72=-T (I973= +T,, and 74= +W Thus, the source function and radiative flux can be written as s(Ty. 7~:70)= kJzo/4~)[J(Ty + 70,Tz,Uo:To)- J(Ty- T,,T,,Uno; To)], (65a) &(TYYTz:To)= zo[Q(Ty+ T,,Tz:go,To)- Q(Ty - T,,, Tz,ao;To)].
(65b)
Two-dimensionalradiative transfer in a finite scatteringplanar medium
593
The variation of the source function and radiative flux across the upper and lower boundaries is presented in Fiis. 6-g. The source function and flux leaving increase as one moves away from the origin. The radiative flux leaving the upper boundary is (l/u) Q(q, 0, a: TV).Near the origin, the medium “sees” the jump in the incident radiation; however, moving away from the origin, the medium itself acts as a shield and reduces-the influence of the jump. At small TV,the source function and the radiative flux approach (l/2) exp (-UT~) and exp (-ao)/Qa), respectively. At large optical distances away from the origin, the medium cannot “see” the jump and the problem is one-dimensional. The distance from the orlgm at which the one-dimensional approximation is valid depends on the optical thickness and the albedo of the medium. Also, since locations on the lower boundary have a better “view” of the
Fig. 6. The sourcefunction for the semi-infinitestep case.
.6
Fig. 7. The influenceof albedo on the sourcefunction for the semi-infinitestep case.
594
A. L. CROSME and J. W. LOEWING
Fig. 8. The radiative flux for the semi-infinitestep case.
jump, larger distances from the origin are required to meet the one-dimensional approximation. As a conservative approximation, the one-dimensional analysis can be applied when ry 2 1.5+ 2~~. The maximum percentage error of this approximation is one per cent. As might be expected, a decrease in albedo decreases the distance. An increase in the optical thickness increases the source function and flux leaving the upper boundary and decreases the source function and flux leaving the lower boundary. #en the optical thickness of the medium is large, the finite medium can be approximated by a semi-infinite medium. The maximum difference between the results for the finite and semiinfinite cases occurs for the one-dimensional case (TV + m). The general behavior of the results for semi-infinite step case follow those for cosine-varying boundary condition if one replaces rY by l/S. Figure 9 illustrates the variation of the reflection function, R(u, ao: TV),with optical distance away from the origin rYfor o = 1.00and 0.90. The reflection function indicates the amount of intensity coming out of the upper boundary of the medium at angle B = se& c which was incident at an angle e, = set-‘uo. The reflection functions for the two albedos exhibit the same general trends of (1) converging to (l/2)(1.0 - exp [-(a + ao)~o]}/(cr+ ao) for small 7Y(0.6% error at TV = 0.01 for r. = 10.0 and o = l.OO), (2) increasing as 7Y increases, and (3) converging to the onedimensional result (madmum percentage error of 0.11 for rY2 20.0, r. zs 10.0 and w I 1.00). The small TV result is obtained by taking TV to zero in Eq. (58c) after the transformation x = /3~~ is made. The reflection functions for o = 0.90 are less than the reflection functions for o = 1.00 because of the absorption in the medium. There is no crossing over effect, as in the transmission function, because only one phenomena, the back scattering, has an effect on the reflection function. Figure 9 also illustrates the variation of the collimated transmission function, T(a; ao: TV), with optical distance away from the origin T,.,for w = 1.00 and 0.90. The transmission function is the amount of incident intensity coming out at the lower boundary, with the attenuation of the beam through the medium subtracted off. The transmission functions for both albedos exhibit the same general trends of (1) converging to (1/2)[exp (-~~o7~)- exp (-cIv~)]/(~ - a,,) for small 7Y(4.8% error at 7Y= 0.01 for 7. = 1.0 and w = 1.OO),(2) increasing as 7Yincreases, and (3) converging to the one-dimensions1 result (maximum percentage error of 0.7 for 7Y2 20.0, 7. d 10.0 and o 5 1.OO). The small TV result is obtained by 'taking TV to zero in Eq. (58d) after the transformation x = /3~~ is made. For a fixed TV, the transmission functions first increase, then reach a maximum near TO = 1.0, and then decrease as the optical thickness of the medium
Two-dimensional radiative transfer in a finite scattering planar medium
tr
595
%
Fig. 9. The reflection and transmission function for the semi-infinite step case.
increases. As the optical thickness increases from zero, more scattering events take place to redirect the intensity from & to 8. As the optical thickness increases further, attenuation becomes important and the transmission function reaches a maximum and then decreases to zero for a semi-infinite medium. The transmission function decreases as the albedo decreases. CONCLUSIONS
When the spatial variation of the incident collimated radiation can be represented by a series of steps, the source function, radiative flux, and intensity can be expressed in terms of those for the semi-infinite step case. This fact greatly reduces the number of calculations. Spatial discontinuities in the incident radiation produce discontinuities in the source function, radiative flux, and intensity. Results for the source function at the boundaries and flux and intensity leaving the boundaries can be obtained without calculating the internal values. For other incident distributions, the source function, radiative flux, and intensity can be calculated from the cosine-varying results. REFERENCES 1. A. L. CROsBIE and T. L. hL%NBARDT, JQS?T 19,257 (1978). 2. A. L. CROSBIE and R. L. DOUGHERTY, JQSRT to, 151(1978). 3. J. H. JOSEPH. W. J. WISCOMBE, and I. A. WEINMAN, J. Atm. Sci. 33,2452 (1976). 4. A. L. CROSBIE and J. W. KOE~~NG, J. Math. AMY. Appl. 57.91 (1977). 5. W. F. BRIZIG and A. L. CROSBIE, JQSRT 14, 1209(1974). 6. W. F. BRIMand A. L. CROSBIE, JQSRT 15, 163(1975). 7. W. F. BIUIIGIUI~ A. L. CROSBIE, IQSRT 13, 1395(1973). 8. J. W. KOE~~NG, Two-dimensional radiative transfer in a finite scattering medium. MS. Thesis, University of MissouriRolls (1974).
INTRODUCTION
AN EXTENSIVE survey (‘) of multidimensional radiative transfer in a scattering medium reveals a lack of exact analyses and results. Recently, two-dimensional radiative transfer in a semiinfinite medium has been investigated. CROSBIEand LINSENBAIU-# studied a rectangular geometry, while CROSBIE and DOUGHERTY’* analyzed a cylindrical geometry. The source function, radiative flux, and intensity at the boundary were expressed in terms of a generalized H-function and were calculated for a wide range of parameters. Two-dimensional effects were most pronounced for conservative scattering. The objective of the present investigation is study of two-dimensional, multiple scattering in a finite medium exposed to collimated radiation. Specifically, the source function, radiative flux, and intensity at the boundaries will be expressed in terms of the generalized X- and Yfunctions. The primary objective is to obtain and present exact numerical results. These “bench mark” solutions should be extremely useful in evaluating various approximate procedures for handling multi-dimensional radiative transfer. PHYSICAL
MODEL
AND GOVERNING
EQUATIONS
The present investigation is concerned with two-dimensional radiative transfer in a finite, absorbing-scattering, planar medium. The medium and coordinate system employed are shown in Fig. 1. The following assumptions are used throughout this study: (1) steady-state intensity and temperature, (2) two-dimensional geometry, (3) no emission, (4) coherent scattering, (5) homogeneous medium, and (6) refractive index of unity. Although monochromatic radiation is considered in the analysis, the frequency subscript has been dropped to simplify the notation. The two-dimensional transport equation can be written as IP(cos @)dp’ d&.
(1).
The radiant intensity, Z, is a function of location (y, z) and direction (p = cosine of the polar angle t9and 4 = azimuthal angle). The extinction coefficient, &, is the sum of the scattering and absorption coefficients, a, + K. The scattering phase function is assumed to be a spike in the forward direction superimposed on an isotropic background, i.e. P(cos 8) =47r@(/.&- /.L’)S(f$- 4’) + I- f, tsupported, in part, by the National Science FoundationthroughGrant ENG 74-22107. SPresentaddress:WestinghouseElectric Corp., West Mifflin, Pennsylvania, U.S.A. 573
(2)
A. L. CROSBIEand J. W. KOEWING
Fig.
I. Definition
of the coordinate system used.
where 8 is the angle between the incident (P’, 4’) and scattered (IL,9) radiation; f is the fraction of radiation scattered in the forward direction; 6 is the Dirac delta function. The quantity f is also equal to the asymmetry factor. The utility of this phase function has been discussed by JOSEPH, WISCOMBE,and WEINMAN.O) The strong forward component is characteristic of the scattering phase function for large particles. Substitution of Eq. (2) into the transport equation and definition of the optical coordinates and albedo as
allows the transport equation to be written as cosfY$+sinBsin$-$+Z=S, .? Y
(4)
where the source function is defined as 2s
S(7yr
72)
=
-g
II 0
+1 I(T~,
T,,
P’,
4’)
dcl’
(5)
W.
-1
Equations (4) and (5) are identical to those for an isotropic scattering medium. For a non-absorbing (conservative) scattering the albedo o is unity, and for a non-scattering medium w is zero and ry = KY and rz = KZ. The medium is exposed to collimated radiation, i.e.
where ~0 is the cosine of the incident polar angle. Note the flux incident on the medium is IO&r,.). The other boundary is not exposed to radiation, i.e. Z-(rP 70,CL,4) = 0. Solution of the transport Eq. (4) and substitution into Eq. (5) yields the following integral equation for the source function:‘4’ ?S(
’ T:; Tyv
ToMTy
-
T:,
Tt -
7;)
dT: dT;,
(7)
where a0 = l/p0 and the kernel is EI(Q 7) = ’s I,- KoU@?j)
dt
(8)
and K. is the modified Bessel function of order zero. The z-component of the radiative flux is given by %(Tyyr
-.
Tz:
TO) =
---
hf(Ty)
eXp
(-c~OT~)/~O+
__
257
‘OS(
Ty,’
7:;
TO)&(Ty
^-
-
T;,
7~ -
7;)
dr: dr;,
(9)
575
Two-dimensionalradiative transfer in a finite scatteringplanar medium
where *
(10)
dy dx.
Ko(xy&%%
The diffuse intensifies leaving the medium in the r,, rz plane are given by
I-( ry, 0, u, ao) = u
z+tq, 70, u, uoo) = fJ I
(11)
S(rY,7:: ro) exp (-UT;) dr:
I0
TO
S(T~,7:: ~~)exp [-u.(T~-
4
(12)
dr:,
0
where u = 11~. In general, f(~~)
can be expressed mathematically with Fourier integral theory as (13)
where g(B) = &j:
(14)
f(rJ exp (-Z&) d7, (D
The source function, z-component of flux, and the intensities leaving the media can be written asg4) ozo -
qAq,
G:
TO) =
-1
ZO
Q,AT,,
UO;
TO) g(B)
ev
(iSTy)
(16)
dS,
I
z77yr
0,
a9 uo)
=
uzo G
m _-coWT
UO:
70) g(B)
exp
MW
WY
(17)
00:
TO) g(B)
exp
WY)
dS,
(18)
I
z+by,
70, WV 00)
=
uzo G
m T&T _ DI
I
JB(7*,
a:
ro)
=
exp (-UT=)+;
QB(rz,
0:
ro)
=
exp (-or&
I0 +T
TO J@(T:, a:
I0
(19)
TO)~,([G - T:),PI dd,
” J@(rL,u:
TO) sgn
(7,
-
WMh
-
T:[,B) dd,
(20)
9 &(a.
uo;
J8(&
TO) = o
uo: TV) exp
(-a;)
d&
(21)
I 0
T&r, uo: 70) = o
I0
” JB(& UO;ro)exp [-u(~~-
721 dr:.
(22)
The generalized exponential integrals are defined as %(r, B) = Imexp (-+r&!(dd@$%,
(23)
exp (-q/m)(dtlt*).
(24)
Physically, the quantities, (oZo/47r>.Ze exp (i/3rY), ZOOSexp Ci/3T,>,(uZ0/47r)RBexp (iPry), and (uZd47r)Tg exp (i/?Ty) are the source function, the radiative flux, the reflected intensity, and the transmitted intensity, respectively, for a medium exposed to cosine-varying collimated radia-
A. L. CROSBIE and J. W. KOEWING
576
tion, i.e. Z&L - pO)S(d) exp (@TV). Inspection of Eqs. (15)-(22) reveals the importance of the function _Z@(T~, a: rO).In the next section, Ambarzumian’s method is used to solve Eq. (19). AMBARZUMIANS
METHOD
Source function
In order to develop the integro-differential equations for the cosine-varying, collimated source function Je at the boundaries of a finite medium, a series of transformations is made. This procedure parallels that of CROSBIEand BREIG.(‘*‘) Let 7z be replaced by 7. - 7z in Eq. (19) to give J@(T~- T,, CT:TV)=
e-"("-~)+
!f 2
I
b"
J8(n-
T:, CT:n) kf,(lr, - 73, B) dr:.
(25)
The key step in the procedure is to treat the optical thickness as a variable. Differentiation of Eqs. (19) and (25) with respect to r. yields ~(~z,~;~O)=~J
( 2
fi
aTO ~(To-w;~o)= aTo
To,
a; 70)%(To-
725PI + y
_ IaaJ,(Tivu;T”~‘&(Tz
T;(,
o
- u eedzo-Q) + T J,(O, a;
TO)
‘0S( +o 2 Io aTo
‘&(IG- 4, P) W
7:, u ; To)
To -
~I(TO-
fj)
dT;.
aTo T,,
(26)
/3)
(27)
At this point in the development, a function QB(rz: ro) is introduced which is defined by
(28) Replacement of
TV
Q(TO
by 7. -
TV
- ~2: TO) =
and 7: by
7. -
T:
; ~%'I(To - ~z,/3)+ ;
gives
I” @s(~o- 7:: ~0)%‘,((T,- ~1,/3) d7:. 0
(29)
Multiplication of Eq. (29) by Jg(~o, u: ro) and comparison of the resulting expression with Eq. (26) yields $@(“’ u; “) = J@(T~.a: ro) @B(~o- TV:TV).
(30)
0
A similar approach applied to Eq. (27) produces
&(70 -
Tz, u:
To)
=- uJ@(~o- rrr a: TO)+ Je(O, a: TO)(PB(~o- ~z: uo).
aTo
(31)
Replacement of u by v(t* + j12) in Eq. (25), multiplication by (o/2) dt/q( t2 + p2), and integration from 1 to 03yields a$(70 - rz: To)=
02
JP(TO Im I
7,
VU2+ ~‘1:~0)$:
B2).
(32)
Setting TV = 0 in Eqs. (30) and (31) and using Eq. (32) produces the following equations for the collimated source function at the boundaries: 2”’
u’
To) = J,(T~,
a:
TO) Q8(To:
To),
0 S(To,
aTo
u;
To)
=
-uJ@(n,, a: TO)+ Ja(O, CT:TO)G$(To: TO).
(33) (34)
511
Two-dimensionalradiative transfer in a finite scatteringplanar medium
Replacement of CLby d/(1 +/3*)/g and x by q( 1 + fi*)/~(t* + fl*) in these equations and the introduction of the generalized X, and Ys functions, X&L; TV)= &(O, g(l + p*)/p: $ and Yh:
70) =
J,~To.
~41
+P*)/P;To),
yields
!z!+; To) = Y&&h: TO)qg(70: To), 370 aY,(cc;To)_ -x41 + p*) 370
-
CL
(35)
Y& : To)+ X&L To)@‘8(70: To),
(36)
where Q/3(70;70)= ; V/(1 + p*) 1’ 0
Y,(x: TO)dx
(37)
xV(l + P2(1 - x2))’
The X, and Ys functions are the two-dimensional analogs of the famous X- and Y-functions of Chandrasekhar. With initial conditions J,(O, cr; 0) = J,(T~, a; 0) = X,&L; 0) = Y&L; 0) = 1, Eqs. (33)-(37) are the basic equations used for evaluating the cosine-varying collimated source functions at the boundaries.
Reflection and transmission functions
In order to relate the reflection and transmission functions to the source functions at the boundaries, a series of transformations must be made. Differentiation of the integral equation for the source function (19) with respect to TVyields
g&T,, 6 70) aTz
= -ueemz +3 Ja(O,a: ro) WT,, p) -t
” !%.I (‘ivu:‘O)$,(lTz +o 2 I0 ar:
_ T;l,
p)
Jg(~o,U: TV)$,(T~ - T,, p) dT;.
(38)
Comparison with Eqs. (28) and (29) gives
aTz
= -u&(Tz,
U:
70)+ J#(o,0:
To)
@‘8(7,:TO)- J,(To,U: To)@@(To-Tz:
To).
(39)
Substitution of u = a0 and multiplication by exp (-orz) and integration form 0 to 7. yields (a + uo)R&~go;To)= Jp(o,co; To)
To @,q(T,;To)e--
dT,
TO @'a(~0- TZ:ro) e--
dT,
1
(40)
with the help of the relation (which is obtained with integration by parts) =
e-c,.ro
&(To,uo:70)- J&h co: To)
+
uR&,
UC,:ro).
(4
Evaluating the equation for the source function (19) at the upper boundary (T*= 0) and interchanging the order of integration yields J,q(T:,
U: To) e-T;V(‘*+B2)dT;$l;
I
@*).
(42)
The inner integral is the reflection function Rp The symmetry of Rp with respect to u and was shown by BREW and CROSBIE.(7) Utilization of the symmetry of RB, interchange
UO
Two-dimensionalradiativetransferin a finite scatteringplanarmedium d( 1 +
579
/3*)/~(t* + p*) leads to
u@o,a:
I
I
xMxY B, 2 0 x+~(1+B2)/a
To)= 1 - 0
- YB(x: ~0) Ja(O,u:
[Xs(x: 70)J,(O, a: 70)- YB(x: TO)&(To, a: TO)]dx,
TO)] dx,
(54)
(55)
where $,(x, p) = (1 + 8*)/U + /3*(1- x2)P2.
(56)
DISCUSSION OF NUMERICAL RESULTS
Cosine-varying
results
The integrodifferential equations (33) and (34) were used to calculate the source functions at the boundaries [&(O, a: TV) and J@(T~, a: TO)], while Eqs. (54) and (55) were used to calculate the fluxes, Qs(O,a: TV) and C&(T~, a: TV), at the boundaries. The numerical procedure is outlined elsewhere.@’ Numerical results are presented in Tables 1 and 2 for a wide range of parameters. A more extensive tabulation is available. ~3)in the discussion that follows, the figures and discussion pertain to normal incidence, u = 1.0. The influences of optical thickness TV,spatial frequency p, and single scattering afbedo w on the source function are illustrated in Figs. 2-4. The source functions for two different albedos exhibit the same general trends; however, the source function for an absorbing medium (o < 1) is less than that for the purely scattering medium (w = 1.0). As might be expected, the influence of albedo is most pronounced at large optical thicknesses and depths and at small spatial frequencies. Except for very large optical thicknesses and albedos near unity, the source function has converged to the one-dimensional result (/3 = 0 for B < 0.01. For 7.5 10.0 and o S 1.0, the maximum percentage error at p = 0.01 is 0.03 for T = 0 and 0.2 for T = TV. As the spatial frequency becomes large, the source function becomes independent of albedo and approaches exp(-or). This result can easily be seen from Eq. (19), since &(T, p) goes to zero as /3+ w At /3 = 10.0, 7. = 10.0 and o = 1.0, the percentage error between the source function and exp(-or) is 0.23 for T = 0 and 4.0 for T = TV.
Fig. 2. The source functionfor the cosine-varyingcase.
Two-dimensional radiative transfer in a finite scattering planar medium
Table 1 (Confd) w=l B
0.3
0.001
T
0
= 0.5
J&bx~,)
J.&,,u;T~)
1.496624 1.496824
1.053783
u
=
1
Qp.a;To)
zx:
01797494 0.797493 0.797492 0.797491 0.797480
0.004 0.008
:%x23 1:053782 1.053775
E% 0:030 0.040
:%%35 11053667
0.797524
W8 ghg
:'m~: I:053320 1.052960
"o%s's:~ 01797614 0.797707
0.002
ym;r;
0.797152
1:4A55R9 1.4774R6 1.467776
sz::t 0:792343
g:
Q,(~J;T,)
xsR:67:gf 01779900 0.772530 0.763121
i&8 0.965742 0*935500
:+gx 1:soo 1.750 2.000
iK25'x 01736994
"o-X%~ 0:859463 0.821572 0.791716 0.767965 0.748130
3-z% 3:500 4.006 4.500 5.000
8=‘73%1 0:700420
t:f%
0.916104 8'XX~os 0:942991 0.940677 0.640408 0.633421 0.620507
:%m(: 1:039071 1.011290 1.026093 1.012375
:%m 1:003913 1.003129 l.CO2606 1.001954
1.000390 1.000260 :':oooo:s~ 1:rooo97 l.OI?OO78
0.631626 0.626521 00%::: 0:618798
0.900545 O.QA3715
~%5550' 0:619311 0.617088
Km 0:991764
8*x533: 0:611695 0.6103R3
KG;260 0:607291
0.607720 0.607481 0.607321 0.607124 0.607005 8'%Kg7 0:606689
8%W 0:606910 0.999667 0.999750 Kz%z 0:999900 "0*:;;9'3; .
0.606561
581
582
A.L.CROSBIE andJ.W. KOEWING
TableI (Contd) w=l
B
J&‘hw,)
T
0
=
1.0
Jg(To.U;~o)
a=1
Q&bJ;To) 0.658671 0.658671 0.658672 0.658673
:‘ws I:757372 1.757368 1.757351 1.757339
Oo%x .
Qg(To.O;~o) 0.658671 0O*txs: 0:658669 0.658663 0.658658 0.658620 0.650556 0.650467
0.659916 0.663557 8%159e)6322 a:665717 :%%5;; 1:546481 1.485017 1.432099 1.387346
0.647817
Ptf25:$ 0:737028 0.554517 0.532950 0.514346 0.864362 0.880189
8't%:i; 0:454709 0.440849 ym;q0
km% 0:404710 0.400556
0.943340 0.94R323
H8%38 01395079 0.391600 0.388903 0.304242
0.377962 0.376420 0.375120
0.980545 0.983715 0.985996 0.907717
0:4156?0 :%%32 1:077707
1.026093 1.022375 1.019584 :-g#i; 1:007e33 1.006265 1.005219 :%%:z 1:003129 1.002606 1.001954 1.001563 1.001302 l.PFO977 1.000781 1.000521 l.COO390
0.369537 0.369328 0.369036
0.997154
0.369366 0.369116 0.36R938 0.3bRROS 8*3%% 0:368341
x%x! 0:999500 0.999667
:%x85 1:000155 :%% .
x%58; 0:377699 0.375631 0.374203
0.999950
00'33%$! 0:369110 0.368064 ix%: 0:367941
Two-dimensional
radiative transfer in a finite scattering planar medium
Table 1 (Contd)
3:801 0.002 :%z 0:oto 0.020 0.030 x40” 0:060 8%8 0:200 0.300 0.400 0.500 0.600 I)%: 1:zso
;454907 :Wo963 _449AS4 ;44se44 .443204 -434058 .422571
0.449073 0.446783
SW%
0: 28704 0.216200 0.159183
:Wdl! .761252 .648909 .545135 :tPW .364126 .296972
3.500 4.000 4.500 5.000 6.000 7.000 8.000 9.000
25.000 30.000 g.38; 60:000 80..000 :x% I so:ooo 175.000 200.000 250.000 300.000
:1:X .190361 .I69914 .I53412 .12R410 :WZ .006227 .077?07 : 8%;:; .044620 .039072 :83K5 .077375 :::%!69: .009795 .007833 :cdoO%~~ .004473 .003913 .003129 : oO%%t .001563 :%g’% .000781 -009521 .000390 :8X% %8&G .000070
0.761456 0.261989 0.262873 8.3% 80’ 0: 267590
oo%% 0:434310 00. y&5 k
8”4m a:540311 0.614646 0.669164 0.718963 0.755573
“6W~f
0:039039 0.024341 0.017696 XlX 1:0514E-02 9.5407E-03 8.9626E-03 8.5800E-03 ;. 3;u&-cg
xm’: 0: 838381
7.-I 279FIO3 7.6425E-03 7.5109F-03
0.924889 0.934764 0.942332
7.4126E-03 7.3365E7.2047F-03 7.1204E-03 7.0619E-03 7.0190E-03 6.9600E-03 6.9215E-03 6.8944E-03 6.R742E-03 6.0779E-03 6.0050E-03 6.7914E-03 6.7806E-03 6.7734E-03 6.7601F-03 6.7645E-03 6.759lE-03 6,7556F6.7512E-03 6.7465E-03 6.7467E-03 6.7445F-03 6.7432F-03 6.7415E-03 6.7406E-03 6.7397E-03 6.7393E-03 6.7390E-03 6.7386F-03 6,7385E-03
03
8*W86” 0:9a77 0.991764 “0*9’83!$Z 0:496Q23 0.9966132 pmg 03
0: 990006 8%KI 0:999001 0.999168 0.999376 on-99Z28V 0:999750 0.999933 0.999875 “o-Z~o3~ 0:999950
17
583
A. L. CROSBIEand
584
J.W. KOEWINC
TableI (Contd) w=l 8
T
0
=
10.0
J6(0.wo)
J&T~AJ;T~)
Qp,o;To)
Q&~o’wTo)
Km3
0.254591
FP ::tz 0:147013
0.146995 0.146992 0.1469'83 0.146948
2: 653174
0.147O66 0.147277
2.651079 2.652695
7.652400 2.650022 2.64bC70 7.640620
I%W’8~ 2: 605303
:*x:i
1:460519 1.410037 1.364126 1.296972 1.250419
%o"o"
6:OOO
7.000 8.000 9.000
:*"0%320~ 1:062309
1.052002
25.OOO 30.000 35.000 40.000
60.000 00.000 E%% 150:000
175.000
:%G%8 l:O31290 3%9833 l:O19504 1.013062 :%m 1:006265 1.('05219 1.004473
5~00-0000s :*:"03;:: 3oo:ooo 1:OO2606 %x008
bOO:OOO 000.000
:‘K%m 1:001302
1.000977 :%%51 l:ooO39o pgx.P~~ 1:000155 :%800tX .
8.
0.
mm;
lt 45808
i? :Et% 0:139799
8%% 0: 125655
"o.mm;;
2.000 2.500 3.000 3.500 4.000
0=1
1:3555E-02 b.Z163E-03 1.4604E-03 4.6981E-04 1.9465E-04 1.2434E-04 9.7615E-05 0.4170E-OS 7.0052F-05 b.4205F-05 b.O39OF-05 5.7017E-05 5.5993E-05 5.4633E-05
5.2744F-05 5.1495E-05 5.0608E-05 4.9946E4.9433E-05 4.8545E-05 4.7977E-05 4.7583E-05 4.7293E-05 yg9g-CJ;
0.173361 0.106750
w$35$ 0:121039
8%~~~: a:429574
0”:83%8 0.017844 0.3024E-03 3.0990E-03 9.7414E-04 3.3173F-O4 1.4744E-04 9.9098E-05
8*f%P8 O:bl4365 0.669109
A.O636F-05
7.1305E-05
%Q% 0:892OSl 0.902721 0.911450
05
4164 2 4Ez05 4.6310E-05
4.6006E-05 4.5052E-05 4.57bOF-05 4.5687G05 4.5639E-05 4.5604E-05 t. ym;-p
4:5519ErO z 4,5489E-05 4.5471F-05 4.5459E-05 4.5444E-05 4.5435E-05 4.5424E-05 4.54 1 BE-05 4.541 ZE-05 4.54096-05 4.5407E-OS 4.5404E-05 4.54035-05
0.962080
0.968133 0.972518 0.975841 0.900545 8%I53X 0:907717 0.991764 0.993805 cm%
0:996682
8%X8 0.999376 0.999500 0.999667
0.999950
b.?391F-05 5.7@49E-05 5.5233F-05 5.3515F-05 5.2303F-05 5. ICOQE-05 5.0159E-05 4.9339E-05 4.0760E-05 4.8328E-05 4.7995E-05 4.7419F-05 4.7052E-05 4.6798E-05 4.66 12E-05 4.6357E-05 4,6190E-05 4.60736-05 4.598bE-05 4.57866-05 4.5688E-05 4.5630E-05 4.5503E-05 *.5553l=-05 4.5531F-05 4.5514E-05 4.5491E-OS 4.547bE-OS 4.5457E-05 4.5445E-05 4.543nF-os 4.5420E-OS 4.5423E-05 4.5415E-05 4.5411E-05 4.5407E-05 4.5406F-05 4.5404E-05 4.5403E-05 4.54026-05
585
Two-dimensionalradiative transfer in a finite scatteringplanar medium
Table 2. Sourcefunction and radiative flux at the boundariesfor the cosine varying case with rO =
I and
(T= I.
w-o B
.99
To
=
1
u = 1
JB(~oq,)
gpdJ;T,)
0.950115 0.950115 0.950114
0.667070 0.667078
WW 0: 950082 00. y9e95 a:94959
3
0.080
8%W8 9:667377 0.667509 0.667842 0.668269
Q&‘aqJ 0.650977 i?6”5oo~T77 0:650976
0.650673 0.650539 0.650199
0.649764
0.600 P-888 I:250
:- WEi 0:765826 0.786689 0.804923
t-%8 2:ooo kZ88 3.500 4.000
4.500 5.000
12.500 Wo"oO
80.000 100.000
125.000
o0wwi 0:980749 0.983884
0.377797 0.37 332 0.37 s 243
0.986141 0.987843
wt3z 0: 370769
00-m% 0: 995086
8:338 #8t 0.3695 0 0.3693 T4 0.369025 “omx 0: 360450
0.999938 0.999950
QSRT Vol. 21, No. 6-G
0.372580 8*33839"2" 0: 369724 0.369351 0.369104
A.L.CROSBIE~II~J. W.KOEWING
Table2 (Cod) w =
Jpd
0.90
PO)
1.597 04 m 8; 1:597 f00 1.597 89 1.597 180 :.g'dg 13596822 1.596606
To
=
1
JB(~Os~:~O)
u
=
1
Qs(O,o;TO)
0.827278 0.827278 0.827277 8.e8tzEE 0:8272SS 8*8e$3& 0:826912 0.826706
Elm l:S9482S
kV358tJ
:.xE l:S62393 1.545427 1.526728
0.7so491 0.757087 0.771386 0.786014
1.326233 ;.m$*~;
0.566512 O.SS1877 8'E8E? o:so315s 0.489033
1:213 !98 ytg;s; 1:1484 z 1 1.134546 '1.most 1:085836
0.420011 0.411401
?8Mf 1:055S41 1.04t429 1.C39884 1.03495s
0.966229 0.971570 yg*g o":31"8%i
:.0at?M 1:020069
1.004694
:.E%f 1:000468
0.922633 0.933961
0".371%% 0:374556
0:982 3 81 0.985406 0.987443 0.988981
8.m: 0:37oso*
fmosi 0:99S538
8.m: .
Oo.%%P% 0:997441
0.368312 0.368203
0.999251 0.999438 iEGm8 0:99977s Oo.z%tB 0:999910 0.999944 0.99995s
00'377%P 0: 76657 3 8.3%% 0:37277C "0*338% .
Two-dimensional
radiative transfer in a finite scattering planar medium
587
Table 2 (Contd) w =
8
J6(0,u;~,)
0.50
T
0
=
J&T~,u;T~)
1
cl=1 Q~hJ;70)
i%OL :‘St:t%
8-885 0:ooe
tii446058
0.446058 0.446058
1:22bLbA
wx25 oo%$ 1:226144 0:030 1.226114
0.040 0.050
0.527492 0.527475 0.527447 0.5374OR
oo%% 0: LOO 0. ZOO
1.000
:-W86’ 1:220958 1.217200 y;l2”6;;
0.515091
1: 196466 1.184800
:%8
1:750 0.433440
6.000
1.058229
T*K% 91000
:‘85ocx 1:040389
“o%z’; 0: 387301 8. g5pfJ;
koO/%S
0:37
::82’:88:
8: GX4fZ 0.375318 0.373795
0.900882
-0.900988 0.901048 0.901544 0.902345 0.9034 00-:8x$ 0:909b 0.913312
18 11
Oo-4?bZ 0:436766 8-WE4 0:423135
“o-X% 0:927LLL 0.931228
:-:x2 0:407565
0.377683 0.375624 p;;w;‘:
3 991
0.9905
1.006467 ::MX 1.003118 1.002599 1.002229
Q&To'UX,)
0.446058
1.226169
19
01372628 8% t “o:t 0:370556 0.370216 “0-366x: 0:368805
0.368844
0.998348
:%.%z 1:001301 1.000976
8*XK 0: 368059 “0*3x5;
0:367951
“0-m:; 0: 367908 0.367897 0.3b7894
:-
3tzo5
033b7902
8-3X8 0:367089
588
A. L. CROSIIIEand J. W. KOEWINC
B Fig. 3.
Fig. 4.
Fig. 3. The influence of albedo on the source function for the cosine-varying case and TO= 1.0, u = 1.0. Fig. 4. The influence of albedo on the source function for the cosine-varying case and 7s = 10.0, u = 1.0.
Under certain conditions, the source function for a finite medium can be approximated by the source function for a semi-infinite medium. The maximum difference between the finite thick and semi-infinite source functions occurs for the one-dimensional case (/I = 0) and for pure scattering (w = 1.00). As the albedo is decreased or the spatial frequency is increased, the difference is drastically reduced. For /3 = 0 and 7. = 10.0, the difference is 0.25 at both boundaries for w = 1.0 and is 0.000022 at the upper boundary (r = 0) and 0.0093 at the lower boundary (T = ro) for o = 0.90. In general, the difference is always smallest at the upper boundary. For the upper boundary, the largest difference is 2.2(IO)-’ for o 5 0.9 and 7. 2 10 and 1.3(10))’ for 7. 2 0.1 and /3~~ z 10.0. The influence of optical thickness, spatial frequency, and single scattering albedo on the radiative flux at the boundaries is illustrated in Fig. 5. Like the source function, the radiative
Fig. 5. The radiative flux for the cosine-varying case.
Two-dimensionalradiative transfer in a finite scatteringplanar medium
589
fluxes leaving the medium decrease as the spatial frequency increases. The radiative flux leaving the upper boundary is (l/a) - (IJO, a; TO). This behavior is simply explained by the fact that the amount of radiant energy per unit time, per unit length incident on the medium in a strip (-Q, 5 TV 5 rb) is 2 sin (&,)/(u/3) and decreases with /3. As would be expected, the flux leaving an absorbing medium (w c 1) is less than that leaving a purely scattering medium (o = 1). When the spatial frequency becomes large, the radiative flux becomes independent of albedo and approaches exp (-UT)/U. An increase in the optical thickness increases the flux leaving the upper boundary and decreases the flux leaving the lower boundary. Semi-infinite step results The numerical results for the semi-infinite step boundary condition, i.e. I
(57)
= i sgn (T& and g(B) = -i/(2,$),
are presented in Tables 3-5. These results were calculated from the following equations:
R(T~, a, uo: TO) =
;[R,d
-
a, uo: To)= -i
T(Tyr
a, uo: TV) sin(B~~)y,
omT&6 00: TO)sin (@TV) $,
(5gd)
S(7y, 7Z;To)= (010/47r)J(Ty, 72,uo: To),
(59a)
I
where
4*(Tyr72: To) I-(~yr
=
0, a, a01 =
ZOQ(Tp
Tzr uo:
To),
@W
Wo/47r) R(~yra, UO;7oh
z+(TyrTara, uo) = (UZol4~)T(Ty,
a,
co;
(59c)
To).
094
The numerical procedure is outlined by KOEWING. (‘) All the results are for normal incidence, uo= 1. This boundary condition is important because other boundary conditions can be expressed in terms of it. The spatial variation of incident radiation can be approximated by a series of steps, i.e. f(ry) = 6 for ?k < ry < Tk+l with k = 1,2,. . . M, where fk is a constant between zero and unity and where 71= --03and T M+l = +Q). This boundary condition can be written as
and the function g(B) becomes
g(B)= (d2@)f =Ifk[-exp (-i/.bk) + exp (-i@Tk+l)].
(61)
Thus, the source function, flux and intensity become S(ryr 7~:70)= (m&,,4a) M fk[J(T,, F=I q*(q,
7r: 70) =
lo
M fk[Q(Tyc =I
Tkr 72, uO:TO)
Tk. Tz, UO: TO)-
Q(Ty
--J(Ty
-
-
Tk+lr Tz,UOo; 70)],
Tk+l, Tz, 00:
To)],
(624
(62b)
A. L. CROSBIEand J. W. KOE~V~NG
590
Table 3.Sourcefunctionandradiativefluxatthe boundariesforthe semi-infmitestepcase witho = and o= 1. T
Y
J(T~.O,~;T~)
J(T~,~,&;T~) T
0.0
0.50000
0.01 0.02
0.50968 0.51591 0.52856 0.54107 0.55427 0.56809 0.57572 0.57860 0.57936
0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
0.57937 0.57937
0
=
Q(T~.O.~;T~)
Q(T~,T~>~;T~)
0.1
0.452423 0.46160~ 0.467623 0.479983 0.49237E 0.50551E 0.520123 0.526943 0.529903 0.53058E 0.53060E 0.530603
00 oo 00 00 00 00 00 00 00 00 00 00
0.50000 0.49482 0.49182 0.48648 0.48234 0.47922 0.47706 0.47641 0.47620 0.47616 0.47616 0.47616
0.452423 0.457373 0.46032E 0.46565E 0.469853 0.47303E 0.475243 0.475903 0.476113 0.476163 0.476163 0.476163
00 DO 00 00 00 00 00 00 00 00 00 00
0.50000 0.49260 0.48739 0.47565 0.46189 0.44408 0.41942 0.40627 0.40024 0.39878 0.39875 0.39875
0.303273 0.30874E 0.312863 0.32268E 0.33499E 0.35203E 0.37720E 0.390973 0.397223 0.39872E 0.39875E 0.39875E
00 00 00 00 00 00 00 00 00 00 00 00
0.50000 0.49188 0.48594 0.47203 0.45469 0.42994 0.30775 0.35661 0.33658 0.32955 0.32934 0.32934 0.32934
0.18394E 00 0.18820E 00 0.19165E 00 0.200433 00 0.212573 00 0.231943 00 0.26996E 00 0.30120E 00 0.321923 00 0.32912E 00 0.329333 00 0.329343 00 0.329343 00
0.50000 0.49107 0.48433 0.46799 0.44662 0.41386 0.34844 0.28305 0.21359 0.14767 0.13209 0.13065 0.13064 0.13064
0.336903-02 0.38178E-02 0.425153-02 0.55242&02 0.75983E-02 0.11648C01 0.232703-01 0.410573-01 0.703073-01 0.114863 00 0.129203 00 0.130633 oc 0.130643 00 0.130643 00
0.50000 O-49099 0.48417 0.46761 0.44585 0.41233 0.34461 0.27543 0.19863 0.11432 0.08237 0.07404 0.07349 0.07350
0.22700E-04 0.131093-03 0.239373-03 0.56402E-03 0.11047E-02 0.218513-02 O.S4149E-02 0.107343-01 0.209363-01 0.455033-01 0.654373-01 0.729583-01 0.734993-01 0.734993-01
T = 0.5 0
0.0 0.01 0.02 0.05
0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
0.50000 0.51306 0.52266 0.54522 0.57327 0.61270 0.67636 0.71843 0.74170 0.74827 0.74841
0.74841
0.303273 00 0.312683 00 0.319973 00 0.337883 00 0.361353 00 0.39618E 00 0.45597E 00 0.49710E 00 0.52020~ 00 0.52675~ oo 0.52689E 00 0.52689E 00 T = 1.0 0
0.0 0.01
0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 20.00 100.00
0.50000 0.51419 0.52492 0.55089 0.58456 0.63498 0.72790 0.80373 0.85746 o.87801 0.87868 0.07869 0.87869
0.183943 0.19111E 0.19700E 0.212243 0.233723 0.269023 0.34290E 0.410693 0.462073 0.482353 0.483033 0.483032 0.48303E
00 00 00 00 00 00 00 00 00 00 00 00 00
T = 5.0 Cl 0.0 0.01 0.02 0.05 0.10 0.20
0.50 1.00 2.00 5.00 10.00 2c.00 50.00 100.00
0.50000 0.51551 0.52757 0.55751 0.59779 0.66137 0.79274 0.92652 1.06730 1.19578 1.22494 1.22760 1.22762 1.22762
0.3369OE-02 0.41175~-02 0.484253-02 0.697293-02 O.l0450E-01 0.17252E-01 0.36847E-01 0.67054E-01 0.11741E 00 0.19635E 00 0.222543 00 0.225173 00 0.225203 00 0.22520E 00 = 10.0 TcJ
0.0
0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 20.00 50.00 100.00
0.50000 0.51565 0.52783 0.55816 0.59910 0.66401 0.79931
0.93960
1.09301 1.25322 1.31088 1.32565 1.32661 1.32661
0.22700E-04 0.208323-03 0.393773-03 0.94983E-03 0.18760~-02 0.372663-02 0.926053-02 O.l8379E-01 0.35892E-01 O.V3309E-01 0.11310E 00 0.12634E 00 0.127303 00 o.12730E 00
I
Two-dimensional
Z3Ty, 0, u9 go) = Wo/49r) M fdR(q c =I
z+(ryr70, uv u0) =
591
radiative transfer in a finite scattering planar medium
(dO/4r)
- Tk,a, uo; 70)- R(q - n+1r a, go;
kz,fk[T(Ty - rk,u:uo:
TO)- T(Ty - 7k+lr&uO:
7011, TO)].
In using the above relations, the following identities are useful: J(-T~, T,, a: 70)= --J(T~,T,, a; 70)and Q(-T~, T,,u: R(-T~. u, uo; TO) = -R(T~,u,
UO;TO)
and
TV) = -Q(T?, T,,U: TV).
T(-T~,u,u~:T~)=
J@s T,, CC70)= 5 J~=o(T,,u: TO)and Q(Q),T,,a:
-T(T~,u,
ro) = i C&,(T~,
NC%u, UO:TO)= i R,&u, uo; TO)and T(m, a, ~00:TV)= i
u~;T~),
U; ro),
Tfico(u,uo: ro).
By evaluating the functions J, Q, R, and T, a wide range of problems can be solved.
Table 4. Source function and radiative flux at the boundaries for the semi-infinite step case with r0 = 1.0 and o= I.
w = 0.99 0.0 0.01 0.02
0.05 0.10 0.20
0.50 1.00 2.00
5.00 10.00 100.00
0.50000 0.51400
0.18394 0.19100
0.52457 0.55012 0.58321 0.63267 0.72353 0.79732 0.84927 0.86897 0.86961 0.86961
0.19678 0.21174 0.23280 0.26735 0.33944 0.40533 0.45499 0.47442 0.47505 0.47506
0.50000 0.49199 o. 48614 0.47244 0.45540 0.43114 0.38994
0.35971 0.34043 0.33374 0.33354 0.33354
0.18394 0.18814 0.19152 0.20015 0.21205 0.23102 0.26813 0.29846 0.31842 0.32528 0.32549 0.32549
w = 0.90
0.0 0.01 0.02 0.05 0.10 0.20
0.50 1.00 2.00 5.00 10.00 100.00
0.50000 0.51234 0.52156 0.54366 0.57193 0.61353 0.68787 0.74583 0.78457 0.79821 0.79860 0.79860
0.18394 0.19004 0.19498 0.20765 0.22527 0.25377 0.31182 0.36303 0..39983 0.41325 0.41364 0.41364
0.50000 0.49292 0.48781 0.47596 0.46142 0.44110 0.40776 0.38457 0.37077 0.36641 0.36630 0.36629
0.18394 0.18759 0.19049 0.19783 0.20783 0.22352 0.25341 0.27677 0.29119 0.29570 0.29581 0.29581
0.50000 0.49643 0.49394 0.48839 0.48191 0.47347 0.46128 0.45435 0.45117 0.45045 0.45044 0.45044
0.18394 0.18568 0.18701 0.19026 0.19449 0.20076 0.21155 0.21871 0.22223 0.22302 0.22303 0.22303
w = 0.50
0.0 0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
0.50000 0.50619 0.51066 0.52102 0.53369 0.55131 0.57982 0.59910 0.60995 0.61302 0.61308 0.61308
0.18394 0.18682 0.18906 0.19459 0.20196 0.21326 0.23430 0.25066 0.26071 0.26368 0.26375 0.26375
@aI (6W
(63~) (63d)
A.L.CROSBIE andJ. W. KOEWING
592
Table 5.Reflection andtransmission function forthesemi-infinite step casewithq,= I andv= I.
T., 0.0 0.01
0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
T 0 = 0.20 R T
0.07520 0.07804 0.07987 0.08363 0.08754 0.09195 0.09729 0.09996 0.10118 0.10147 0.10147 0.10147
0.07421 0.07702 0.07884 0.08259 0.08648 0.09089 0.09622 0.09889 0.1001; 0.10040 0.10041 0.10041
.r = 0.50 cl
T0 = 1.00
T = 2.00 0
R
R
T
R
T
0.15837 0.16717 0.17379 0.18968 0.21012 0.24035 0.29495 0.33844 0.36870 0.38014 0.38051 0.38051
0.11627 0.12330 0.12872 0.14204 0.15964 0.18653 0.23726 0.27927 0.30914 0.32054 0.32091 0.32091
0.16625 0.17648 0.18440 0.20401 0.23026 0.27134 0.35414 0.43340 0.50550 0.54836 0.55153 0.55157
0.05851 0.06328 0.06723 0.07758 0.09247 0.11775 0.17548 0.23895 0.30353 0.34491 0.34807 0.34811
0.16705 0.17356 0.18919 0.20925 0.23888 0.29218 0.33441 0.36361 0.37454 0.37490 0.37490
0.12320 0.12853 0.14161 0.15888 0.18521 0.23470 0.27549 0.30430 0.31519 0.31555 0.31555
0.17630 0.18407 0.20326 0.22888 0.26885 0.34896 0.42496 0.49328 0.53305 0.53588 0.53591
0.06317 0.06702 0.07709 0.09153 0.11598 0.17151 0.23212 0.29314 0.33147 0.33429 0.33432
T
o .12g48
0.11933
0.13568 0.14010 0.15020 0.16227 0.17850 0.20322 0.21883 0.22729 0.22966 0.22971 0.22971
0.12518 0.12939 0.13908 0.15078 0.16666 0.19112 0.20668 0.21513 0.21749 0.21754 0.21754 w = 0.99
0.01
0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
0.07801 0.07981 0.08354 0.08739 0.09175 o.og700 0.09963 0.10083 0.10111 0.10112 0.10112
0.07699 0.07878 0.08249 0.08633 0.09068 0.09593 0.09857 0.09976 0.10005 0.10005 0.10005
0.13560 0.13997 0.14992 0.16182 0.17778 0.20204 0.21732 0.22557 0.22787 0.22792 0.22792
0.12511 0.12526 0.13882 0.15034 0.16596 0.18997
0.20519 0.21343 0.21572 0.21577 0.21577 w = 0.90
0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
0.07773 0.07935 0.08267 0.08609 0.08992 0.09450 0.09677 0.09780 0.09804 0.09804 0.09804
0.07671 0.07832 0.08163 0.08504 0.08887 0.09344 0.09571 0.09673 0.09698 0.09698 0.09698
0.13494
0.13880 0.14755 0.15790 0.17162 0.19206 0.20459 0.21117 0.21295 0.21299 0.21299
0.12448 0.12815 0.13654 0.14656 O.i5998 0.18019 0.19267 0.19924 0.20102 0.20106 0.20106
0.16599 0.12232 0.17487 0.06235 0.17165 0.18508 0.20208 0.22674 0.26975 0.30234 0.32367 0.33107 0.33128 0.33128
0.12692 0.13809 0.15263 0.17441 0.21414 0.24551 0.26651 0.27388 0.27409 0.27409
0.18143 0.19737 0.21823 0.24992 0.31053 0.36412 0.40785 0.42946 0.43060 0.43061
0.06546 0.07344 0.08464 0.10312 0.14331 0.18455 0.22264 0.24320 0.24433 0.24434
0.16214 0.16483 0.17098 0.17836 0.18838 0.20399 0.21406 0.21954 0.22105 0.22108 0.22108
0.11922 0.12136 0.12636 0.13254 0.14120 0.15535 0.16492 0.17027 0.17177 0.17180 0.17180
0.17035 0.17332 0.18018 0.18859 0.20035 0.21973 0.23350 0.24196 0.24468 0.24476 0.24476
0.06018 0.06144 0.06451 0.06852 0.07456 0.08591 0.09541 0.10218 0.10463 0.10470 0.10470
w = 0.50 0.01 0.02 0.05 0.10 0.20 0.50 1.00 2.00 5.00 10.00 100.00
0.07655 0.07740 0.07911 0.08082 0.08269 0.08482 0.08593 0.08626 0.08637 0.08637 0.08637
0.07554 0.07639 0.07809 0.07980 0.08166 0.08379 0.08480 0.08524 0.08534 0.08534 0.08534
0.13230 0.13423 0.13849 0.14333 0.14945 0.15786 0.16251 0.16472 0.16526 0.16527 0.16527
0.12197 0.12381 0.12788 0.13256 0.13852 0.14682 0.15145 0.15366 0.15420 0.15421 0.15421
For example, the case of a strip of collimated radiation, i.e.
can be expressed in terms of the present investigation as follows: M = 3, f, = fJ = 0, f2 = 1, 71=-m, 72=-T (I973= +T,, and 74= +W Thus, the source function and radiative flux can be written as s(Ty. 7~:70)= kJzo/4~)[J(Ty + 70,Tz,Uo:To)- J(Ty- T,,T,,Uno; To)], (65a) &(TYYTz:To)= zo[Q(Ty+ T,,Tz:go,To)- Q(Ty - T,,, Tz,ao;To)].
(65b)
Two-dimensionalradiative transfer in a finite scatteringplanar medium
593
The variation of the source function and radiative flux across the upper and lower boundaries is presented in Fiis. 6-g. The source function and flux leaving increase as one moves away from the origin. The radiative flux leaving the upper boundary is (l/u) Q(q, 0, a: TV).Near the origin, the medium “sees” the jump in the incident radiation; however, moving away from the origin, the medium itself acts as a shield and reduces-the influence of the jump. At small TV,the source function and the radiative flux approach (l/2) exp (-UT~) and exp (-ao)/Qa), respectively. At large optical distances away from the origin, the medium cannot “see” the jump and the problem is one-dimensional. The distance from the orlgm at which the one-dimensional approximation is valid depends on the optical thickness and the albedo of the medium. Also, since locations on the lower boundary have a better “view” of the
Fig. 6. The sourcefunction for the semi-infinitestep case.
.6
Fig. 7. The influenceof albedo on the sourcefunction for the semi-infinitestep case.
594
A. L. CROSME and J. W. LOEWING
Fig. 8. The radiative flux for the semi-infinitestep case.
jump, larger distances from the origin are required to meet the one-dimensional approximation. As a conservative approximation, the one-dimensional analysis can be applied when ry 2 1.5+ 2~~. The maximum percentage error of this approximation is one per cent. As might be expected, a decrease in albedo decreases the distance. An increase in the optical thickness increases the source function and flux leaving the upper boundary and decreases the source function and flux leaving the lower boundary. #en the optical thickness of the medium is large, the finite medium can be approximated by a semi-infinite medium. The maximum difference between the results for the finite and semiinfinite cases occurs for the one-dimensional case (TV + m). The general behavior of the results for semi-infinite step case follow those for cosine-varying boundary condition if one replaces rY by l/S. Figure 9 illustrates the variation of the reflection function, R(u, ao: TV),with optical distance away from the origin rYfor o = 1.00and 0.90. The reflection function indicates the amount of intensity coming out of the upper boundary of the medium at angle B = se& c which was incident at an angle e, = set-‘uo. The reflection functions for the two albedos exhibit the same general trends of (1) converging to (l/2)(1.0 - exp [-(a + ao)~o]}/(cr+ ao) for small 7Y(0.6% error at TV = 0.01 for r. = 10.0 and o = l.OO), (2) increasing as 7Y increases, and (3) converging to the onedimensional result (madmum percentage error of 0.11 for rY2 20.0, r. zs 10.0 and w I 1.00). The small TV result is obtained by taking TV to zero in Eq. (58c) after the transformation x = /3~~ is made. The reflection functions for o = 0.90 are less than the reflection functions for o = 1.00 because of the absorption in the medium. There is no crossing over effect, as in the transmission function, because only one phenomena, the back scattering, has an effect on the reflection function. Figure 9 also illustrates the variation of the collimated transmission function, T(a; ao: TV), with optical distance away from the origin T,.,for w = 1.00 and 0.90. The transmission function is the amount of incident intensity coming out at the lower boundary, with the attenuation of the beam through the medium subtracted off. The transmission functions for both albedos exhibit the same general trends of (1) converging to (1/2)[exp (-~~o7~)- exp (-cIv~)]/(~ - a,,) for small 7Y(4.8% error at 7Y= 0.01 for 7. = 1.0 and w = 1.OO),(2) increasing as 7Yincreases, and (3) converging to the one-dimensions1 result (maximum percentage error of 0.7 for 7Y2 20.0, 7. d 10.0 and o 5 1.OO). The small TV result is obtained by 'taking TV to zero in Eq. (58d) after the transformation x = /3~~ is made. For a fixed TV, the transmission functions first increase, then reach a maximum near TO = 1.0, and then decrease as the optical thickness of the medium
Two-dimensional radiative transfer in a finite scattering planar medium
tr
595
%
Fig. 9. The reflection and transmission function for the semi-infinite step case.
increases. As the optical thickness increases from zero, more scattering events take place to redirect the intensity from & to 8. As the optical thickness increases further, attenuation becomes important and the transmission function reaches a maximum and then decreases to zero for a semi-infinite medium. The transmission function decreases as the albedo decreases. CONCLUSIONS
When the spatial variation of the incident collimated radiation can be represented by a series of steps, the source function, radiative flux, and intensity can be expressed in terms of those for the semi-infinite step case. This fact greatly reduces the number of calculations. Spatial discontinuities in the incident radiation produce discontinuities in the source function, radiative flux, and intensity. Results for the source function at the boundaries and flux and intensity leaving the boundaries can be obtained without calculating the internal values. For other incident distributions, the source function, radiative flux, and intensity can be calculated from the cosine-varying results. REFERENCES 1. A. L. CROsBIE and T. L. hL%NBARDT, JQS?T 19,257 (1978). 2. A. L. CROSBIE and R. L. DOUGHERTY, JQSRT to, 151(1978). 3. J. H. JOSEPH. W. J. WISCOMBE, and I. A. WEINMAN, J. Atm. Sci. 33,2452 (1976). 4. A. L. CROSBIE and J. W. KOE~~NG, J. Math. AMY. Appl. 57.91 (1977). 5. W. F. BRIZIG and A. L. CROSBIE, JQSRT 14, 1209(1974). 6. W. F. BRIMand A. L. CROSBIE, JQSRT 15, 163(1975). 7. W. F. BIUIIGIUI~ A. L. CROSBIE, IQSRT 13, 1395(1973). 8. J. W. KOE~~NG, Two-dimensional radiative transfer in a finite scattering medium. MS. Thesis, University of MissouriRolls (1974).