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Grey radiative transfer
J. Quant. Spectrosc. Radiat. Transfer. Vol. 1I, pp. 597-615. Pergamon Press 1971. Printed in Great Britain
GREY
RADIATIVE
TRANSFER
G. C. POMRANING Science Applications, Inc., P.O. Box 2351, La Jolla, California 92037
Abstract--The general problem of reducing a frequency-dependent radiative transfer problem to an equivalent grey problem is discussed. In particular, we construct, from the integral transport equation, an asymptotic solution to be used in forming grey opacities. This is closelyrelated, but not identical, to the so-calledB-N method widely used in neutron transport work. This leads to generalizations, to non-zero temperature gradients, of the standard Planck and Rosseland mean opacities. One also obtains the mean opacities relevant to higher angular moments of the equation of transfer. We also show how the angular dependence of this asymptotic result can be incorporated into the grey equation of transfer without introducing an angularly dependent opacity. One needs only to modify the scattering kernel. The dominant term in the modification depends upon the differencebetween the Planck and Rosseland mean opacities. Our resulting grey equation is shown to be exact in the two limiting cases of optically thin and thick systems where it is well known that the standard Planck and Rosseland means, respectively, are the appropriate averages. Numerical results are presented for idealized absorption coefficients, such as the Elsasser band (an infinite array of equally spaced, equal strength Lorentz lines) and a 1/v a behavior (characteristic of bound-free and free-free transitions). We calculate the generalized Planck and Rosseland means and exhibit their dependence upon the temperature gradient. Results from our grey equation of transfer are compared with an exact frequency-dependent result for a model problem and shown to be significantly more accurate than those predicted by the usual grey models employing the standard Planck and Rosseland means. All of our considerations apply to the formation of a multigroup equation of transfer as well as to a grey, or one group, equation.
I. I N T R O D U C T I O N IN PRACTICALapplications of the theory of radiative transfer, a c o m m o n simplification is to use a grey, or o n e group, e q u a t i o n of transfer rather t h a n the correct m u l t i f r e q u e n c y equation. This involves the use of a b s o r p t i o n coefficients averaged over frequency. The usual procedure is to e m p l o y a single m e a n a b s o r p t i o n coefficient in both the emission a n d a b s o r p t i o n terms. This m e a n is generally t a k e n as the P l a n c k mean, a n average of the a b s o r p t i o n coefficient over the P l a n c k function, or the R o s s e l a n d mean, a n average of the m e a n free path over the t e m p e r a t u r e derivative of the P l a n c k function. It is well k n o w n that such grey e q u a t i o n s are only correct in limiting cases. The P l a n c k m e a n is a p p r o p r i a t e in the optically thin, emission d o m i n a t e d limit, a n d the R o s s e l a n d m e a n is correct in the optically thick, a l m o s t u n i f o r m t e m p e r a t u r e limit. I n this p a p e r we propose a new grey e q u a t i o n to bridge the gap between these two limiting cases. The analysis involves two distinct steps: (1) the i n t r o d u c t i o n of a n g u l a r effects, which are necessarily present in a grey e q u a t i o n , w i t h o u t i n t r o d u c i n g a n a n g u l a r l y d e p e n d e n t opacity; a n d (2) the calculation of a n a s y m p t o t i c s p e c t r u m to use as a weight f u n c t i o n in f o r m i n g grey opacities. O u r ideas are similar to those found in the n e u t r o n t r a n s p o r t theory literature. The success of this related a p p r o a c h in that field p r o m p t e d the present investigation in the radiative transfer context. 597
598
G . C . POMRANING
We apply the results of our analysis to two idealized absorption coefficients, namely the Elsasser band representing a pure line spectrum, and the 1Iv 3 continuum cross section. We also compare results from our grey equation, as well as results from the usual Planck and Rosseland grey equations, with an exact frequency dependent calculation involving the Elsasser band. The new grey equation is, at least for this model problem, more accurate than the Planck and Rosseland grey equations. It should be pointed out that the analysis given here applies to the development of a multigroup equation as well. In fact, our approximations are more valid in the multigroup case than they are in the grey, or one group, case. We chose to work in this paper with the grey case for simplicity of presentation, as well as the recognition that grey equations of transfer are widely used in practice. 2. R E V I E W O F T H E G R E Y M E T H O D
The details of the development of our grey equation of transfer have recently been given3 t) For completeness in the present paper, we summarize the analysis, omitting much of the algebraic detail. With the neglect of scattering, the time-independent equation of transfer in a planar atmosphere is
#
cgl(z, v, #) - a(z, v)[B(T, v ) - l(z, v,/0], ~z
(1)
where z, v, p, and T are the standard space, frequency, angle (measured with respect to the positive z axis), and temperature variables, I(z, v, I~) is the specific intensity of radiation, a(z, v) is the absorption coefficient suitably corrected for induced emission, and B(T, v) is the Planck function if local thermodynamic equilibrium is assumed. Integration of equation (1) over all frequency yields
#
(~I(z,#) = ap(z)~T'(z)--~(z, 12)l(z,#), dz
(2)
where ~ is a constant given by
7 = 2~4k4/15c2h 3,
(3)
and I(z, iz) is the frequency independent, or grey, specific intensity defined as
l(z, lO = r dvl(z, v, #).
(4)
o
Equation (2) also contains two mean interaction coefficients, namely the Planck mean
ae(z) =
~ dva(z, v)B(T, v) ~o dvB(T, v) '
(5)
and
~ dza(z, v)l(z, v, 12) ~(z,/~) =
~o dvI(z, v,/~)
(6)
Grey radiative transfer
599
The frequency independent equation of transfer, equation (2), is exact. However, to compute 6(z, p) one needs know l(z, v, ~) which, of course, is presumed unknown. Hence, in practice one must replace I(z, v, #) in equation (6) by a weight function, i(z, v, p), which should be an approximation in some sense to l(z, v, #). The entire question of what is an appropriate weight function constitutes the grey opacity problem. We shall return to this point shortly. At present, however, we note that by introducing two grey opacities in equation (2), namely trp and 6, we have assured ourselves of the correct limiting behavior for an optically thin system. In this case, l(z, #) is much less than 7T 4 and hence the choice made for 6 is irrelevant. The only important thing is that the emission coefficient be given by the Planck mean, and this is the case in equation (2) no matter what choice is made for 6. However, we note in general that 6 depends upon angle which detracts from the appeal of equation (2) as a grey equation of transfer. Accordingly, we consider an alternate scheme. As an intermediate step in the development, we expand the specific intensity in Legendre polynomials according to
l(z, v, I-O = . ~= o ~2n + 1 I . ( z ,
(7)
v)P.(#).
Equation (I) is then equivalent to an infinite set of equations for the expansion coefficients
I,(z, v), the nth equation being of the form n
0I._ x(z, v)
Oz
~-(2n+ 1)a(z, v)l.(z, v)+(n+ 1)
,91.+ 1(_7, v)
~3z
= 0.
(8)
If we define the grey moments as
I.(z) = f dvI.(z, v),
(9)
integration of equation (8) over all frequency yields the grey moment equation 01. 1(Z) l-(2n + 1)a.(z)l.(z) + (n + 1) 0_~,)I_(z= O. GZ 0z 1
n - -
(1 O)
Here a.(z) is given by
o.(z) =
5~ dva(z, v)t.(z, v) S~ dvI.(z, v)
(1 I)
As before, since the I,,(z, v) are not known one must use an approximation, say I,(z, v), in equation (1 I) to compute the moment opacities. Now, since equation (7) represents a convergent expansion, the I,,(z, v) and hence the I,,(z) are arbitrarily small for n large enough. Hence, for large n, say n > N, the coefficient of I,,(z) in equation (10), namely a,,(z), is unimportant. That is, as long as a,,(z) is a number of reasonable magnitude, its precise value is irrelevant, for large enough n. In view of this, we replace a,,(z) in equation (10) by tr*(z) for n > N. That is, for n > N we use a common
600
G, C. POMRANING
grey opacity, independent of n. With this replacement, the infinite set of grey moment equations are entirely equivalent to the grey equation of transfer
OI(z, la) ap(Z)TT4(z)- a*(z)l(z, #)+ f d~'ac(z, ~ . ~')l(z, I~'). I~ 8z =
(12)
4n
Here
ae(z, ~ . ~')
is an effective scattering kernel given by
ae(z, , . ,') = f ~ n l [cr*(z)-a.(z)]P,(f~ . ,').
(13)
n:O
In view of the arguments just given, it is evident that tr*(z) is completely arbitrary if N is large enough. In practice, however, N is likely to be chosen as a quite small integer, so the choice of tr*(z) is of some importance. We have considered two choices, namely a*(z) = Cro(Z),
(14)
a*(z) = aN(z).
(15)
and
The supporting arguments for these choices are made in our earlier paper. (1) It is also easily argued that, no matter what choice is made for a*(z), equation (12) is correct in the optically thin limit since the erhission mean is the proper Planck average. As a simple example of equation (12), let us set N = 1. Then, if we choose a* = ao, equation (12) gives OI(z, p) = 0"p(Z)~ T 4 ( z ) -- O'o(z)l(z ,/2)
1
+~ #[ao(Z)-al(z)] f d.'ll'I(z,l~').
(16)
-1
On the other hand, setting a* -- a 1, we find equation (12) yields, with N --- 1, 1
OI(z, I~) _ ap(Z)TT4(z)--al(z)I(z, !a)+½lax(z)--ao(Z)] f d!a'I(z, la'). Oz
(17)
-1
Our numerical results in Section 4 indicate that both equations (16) and (17) are more accurate than the usual Planck and Rosseland grey equations, with equation (17) being more accurate than equation (16). It must also be pointed out that the multifrequency equation given by equation (1) contains no scattering, whereas our grey form of the same equation, namely equation (12) or the special cases equations (16) and (17), contain scattering-like terms. These terms make
Grey radiative transfer
601
our grey equation much more difficult to solve than the usual grey equations. However, had equation (1) contained scattering terms, the additional scattering-like terms introduced in forming the grey equation would only serve to modify the physically meaningful scattering kernel. Hence, in this more general case of radiative transfer with scattering, our grey equation is no more complex in form than the usual, but less accurate, Planck and Rosseland grey equations. To complete the specification of the grey equation of transfer, we must compute the moment opacities tT, needed in equation (13). These quantities are well defined by equation (11) and our only task is to obtain an approximation to the I,(z, v) for use in this equation. To this end, we seek an asymptotic solution of equation (1) which yields a generalization, to non-zero temperature gradients, of the equilibrium diffusion theory result for l(z, v, I~). In terms of optical depth z, defined as, 2
T
= I dz'a(z', v),
(18)
p/
0
the asymptotic solution of equation (1) is
I(z, V,l~) = [l +-~lB(z),
(19)
where the operator D is defined by D = 0/0z.
~
(20)
The operator appearing in the denominator of equation (19) is merely a shorthand notation for a Taylor series expansion about/~D = 0. Equation (19) was derived by writing equation (1) in integral form, expanding B(z') appearing in the integrand in a Taylor series about z' = z, and integrating term by term, ignoring edge effects. Equation (19) can be rewritten as ],/202
I(z, v,#) = ( a - # D )
q
1 -~ l + ~ 2 - J B ( z ) .
(21)
The common equilibrium diffusion theory used in astrophysics corresponds to neglecting kt2D2/(1-/~2D2) as compared to unity in equation (21). Physically this means that the temperature is a sufficiently weak function of optical depth that one can neglect its second and higher derivatives. In the present treatment we assume, for the purposes of dealing with this term, that
D2B(z)
= 0~2B(z).
(22)
That is, we assume, in the vicinity of z, that B(z) can be represented by an arbitrary linear combination of exponentials (0c2> 0) or trigonometric functions (ct2< 0), and hence satisfies a Helmholtz equation. If we further assume that 0~2 is a weak function of space, then use of the successively differentiated results from equation (22) in equation (21) gives I(z,v,#)=
I 12~2~(1 1--/t
j
-#D)B(z).
(23)
602
G.C. POMRANING
The special case 0t = 0 corresponds to equilibrium diffusion theory. Reintroducing the variable z in favor of z, equation (23) can be rewritten as
I(z, v, p) = ~
fi dz 8-T]
v).
(24)
Equation (24) is the asymptotic spectrum we sought, except that we need specify ct, which in general is a function of space and frequency, although the spatial dependence has been assumed weak. From equation (22), assuming fi is a slowly varying function of space (this is necessary, as we shall see, ifa is to be a weak function of z), we find
(l ~B~d2T ll(~2BI[dWI2
(25)
Now, this entire paper has presumed that fi(z, v) is a strong function of v, due to lines, etc., since otherwise the question of how to form a grey equation of transfer is not a significant one. Accordingly, we assume that the frequency variation of a in equation (25) comes from fi, not B. We then write 0"2~ 2 =
constant in frequency
(26)
=/~2,
and the asymptotic spectrum, equation (24), becomes
I(z,v,#)=
1__/2~82/fi 2
B(T,v).
1 f dz
(27)
Finally, we determine e by using equation (26) in equation (22) and integrating over all frequency. The result is 1 ~2T4 /~2 - - T 4
(28)
~z 2 .
If we consider the asymptotic result, equation (27), as the approximate intensity representation 7(z, v, #) to be used to compute the moment opacities, then the corresponding angular moments needed in equation (l 1) are 1
1
72,(z,v)=2rcB(T,v)fd#P2n(#)(l_#~2/fi2 ),
(29)
-1 1
f
i2.+I(Z,V)= --2~ dr dz OB(T,v) or f1 - !
dgP2.+l(~)
(1--~8' )
E/fiE ~"
(3(I)
If we define 1
An(C) =
~
d~Pn(/~) --1
(31)
Grey radiative transfer
603
then equations (11), (29), and (30) yield f~o dYO'(Z,Y)Wn(T , v)hn(,~/~7) ~ dvw.(T, v)A.(e/a) '
a.(z) =
(32)
where w.(T, v)
~ B(T, v) for n even,
(33)
= (OB(T, v)/OT for n odd. In particular, 1
.
Ao(~) : ~ I n / ~ ] '
(34)
and the other A.(¢) follow from the recurrence relation A.+ ~(¢) = / n-~-]-]~ A.(~)- ~
._,(~)---~ 6.o.
(35)
If e is pure imaginary (e2 < 0) then we set e = i6 and define the real quantity A.(rl) = i"A.(irl).
(36)
In this case a.(z) =
~ dvtr(z, v)w.(T, v)A.(6/a) ~ dvw.(T, v)A.(6/a)
(37)
Specifically, we have 1 .4o(r/) = - t a n - ~(r/), t/
(38)
and the recurrence relation -
n
_
,+
1
,39,
for n > 0. Finally, in the limit of small e or 6, we obtain
So dvw.(W, a, = S~ dvw.(T, v)a-" "
(40)
In particular, in this limit a o in the Planck mean and al is the Rosseland mean. Hence ao and al can be considered as generalizations, to non-zero temperature gradients, of the Planck and Rosseland means, respectively. Because of the limiting behavior of tr, to the Rosseland mean, one can easily show that our grey equation of transfer, equation (12), upon making the diffusion approximation, will give the correct equilibrium diffusion theory relationship, i.e. the mean absorption coefficient involved will be the Rosseland mean. As we noted earlier, equation (12) is also correct in the optically thin, emission dominated, case since the emission coefficient is always the Planck mean.
604
G . C . POMRANING 3. A P P L I C A T I O N
TO IDEALIZED
ABSORPTION
COEFFICIENTS
As an example of the use of the asymptotic spectrum to compute the moment opacities, let us consider an infinite array of equally spaced, equal strength, Lorentz lines. This leads to the Elsasser absorption coefficient S
sinh(2nh/d)
a(v) = d [cosh(2nh/d)-cos(2nv/d)]"
(41)
Here S and h are the total area and halfwidth at half maximum, respectively, of a single line, and d is the spacing between lines. We choose a distance scale such that the minimum value of a(v) over v is unity. Then cosh fl + l a(v) = cosh f l - cos(2nv/d)' (42) where fl = 2nh/d. We further assume the spacing d to be small enough so that B and aB/OT are in effect constant in frequency over a distance d. Then, for e = 0, in which case equation (40) is applicable, we find 1)~. dx(cosh f l - c o s x)"- x a.(e = 0) = (cosh fl + J'~ dx(cosh f l - cos x)" " (43) This leads to
ao(e=O)=at,-
cosh fl + 1 sinhfl '
(44)
and a.(e=0)=
[cosh fl + l i p ._ l(coth fl) [ sinhfl ] P.(cothfl) "
(45)
In particular, equations (44) and (45) give the ratio of Planck to Rossetand means as a0(e = 0) = __at,= coth fl, ax(e = 0)
(46)
at
a well known result for the Elsasser band. Figures 1 through 4 show typical results for a o and a 1 for a general temperature gradient (e2 not necessarily small) corresponding to the Elsasser band, as calculated from equations (32) and (37) for n = 0 and 1. These integrals can be carried out analytically, but the resulting expressions are sufficiently complex that we found it more convenient to perform the integrations numerically. These results for ao and al can be considered as generalizations, to non-zero temperature gradients, of the Planck and Rosseland means. The quantities actually plotted in these figures are the ratio of a o to the Planck mean, and the ratio of trl to the Rosseland mean, i.e. ao at,
ao a0(e = 0)
sinh fl cosh fl + 1 ao,
(47)
al
al
aR
ao(e = 0)
cosh fl cosh fl + 1 at.
(48)
and
Grey radiative transfer
605
0-9~:
# b
o
O-S
0
012
0.14
0.16
018
~2
FIG. 1. Elsasser band, ao for positive e2.
It is interesting to note that for e 2 > 0, both ao and ~1 are less than the Planck and Rosseland means, respectively, whereas for ~2 < 0, go and al are greater than gp and trR. This behavior can be shown to be true for any a b s o r p t i o n coefficient. We also note that Ol, the generalized Rosseland mean, is a somewhat stronger function o f ~2 than is o-o.
1"6
1"4 b~
1.2I
1.0I0'01
0"I
li-O
FIG. 2. Elsasser band, cro for negative ~2.
IO-0
606
G . C . POMRANING 1,0
0"95
b=
0"90
b-
02
0.4
0'6
0'8
~2
FIG. 3. Elsasser band, al for positive e 2.
One can obtain relatively simple analytic results in limiting cases. In general, as 32 = - e 2 approaches infinity, equation (37) gives, for n = 0 and 1,
f~ dva2(v)B(T, v) ao = f~ dvtr(v)B(T, v ) '
(49)
f~ dvtr2(v)tgB(T, v)/aT at = f~ dva(v)~B(T, v)/t~T"
(50)
16
1.4
b1.2
~'0 0.01
0.I
1.0 _E 2
FIG. 4. Elsasser band, e I for negative e 2.
I0.0
Grey radiative transfer
607
For the Elsasser band, this leads to O"o
--
' c o t h fl,
(51)
0"1 ~_.g coth2fl.
(52)
tTp
and aR
On the other hand, for small •2 one obtains for the Elsasser band
0"0 = l_soe2+O(e4),
(53)
Gp
and
a-!l = 1 - sle 2 + O(e'*),
(54)
O"R
where
_1F 2 + e -2" -] So = 6L(cos h/~+ 1 ) 2 j ,
(55)
3 Sl - 5(cosh fl+ 1) 2.
(56)
and
In the limit of fl = 0, one can obtain the entire e 2 dependence of these moment opacities in simple analytic form. The results for 0"0 are 0"0 = e ap ~/2[1 - x/(1 -e2)] 1/2'
(57)
or, if 6 2 = _ g 2 ,
ao 0"e
6 4 2 1 4 ( i +62) -
i]i/z
(58)
.
Similarly, for cr~ we obtain 0"1
0[(8 + 3e 2) [1 - x/(1 -- e2)] ,/2 _ 4el1 + ~/(1 - e2)] l/z]
-j.
O"R
(59)
and O"I
1 F4~%/(I '-I- 62)-'~ - 1] I/2 - - ( 8 -
Str - 10 L-
3~2)['4(1 '-l'-62) - 111/2q
~-v~i~i~~+-~-2~]]
i7-2
_]
(60)
As a second example, consider the simple 1/v 3 absorption coefficient, i.e.
a(v) = # { h~To)- a (1--e-h~/kr ).
(61)
608
G . C . POMRANING
The exponential terms in equation (61) constitute the correction for induced emission. We choose a distance scale such that #(To~T) 3 = 1. T h e n o(v) = x - 3(1 - e-X),
(62)
where x = hv/kT, and the generalized Planck and Rosseland means are, from equation (37) with n = 0 and 1,
(x3)
J'~ dxa2(x) ~
t a n - l(6/a)
j'~ dxa(x) ~
tan-l(6/o)
ao =
,
(63)
and 2
I- x'*e" -II-o
_
[- x'ie x -]]-tr
_
.i'~° dx. (x)[~_lL~tan {71 =
s: dx.(x)[
/l
]
'(6/~r)-I
tan '(6/.)-
(64)
1]"
F o r 6 = 0, equations (63) and (64) yield (65)
ao = av = 15/n* = 0-1540, ~4
a~ = aR =
(66)
= 0-005089. 9450 ~
[2+11
.=1 t,, 6
,,7#
F o r small values o f 6 2, equations (53) and (54) give the 62 (or e 2) dependence with s o = 1844 and st = 657,000. Numerical results for other values of 62 are shown in Figs. 5 and 6.
i-6
i'4
IC iO'e
;0-4
fO-5 _E 2
FIG. 5. Continuous absorption, tro for negative e 2
I0"3
Grey radiative transfer
609
1.4
-%
E
i
I-0 10"7
i 0 -a
10 "6
i 0 +$
FIG. 6. Continuous absorption, a I for negativee2. Comparison of these results with those for the Elsasser band, Figs. 2 and 4, shows that the curves are very similar in nature. This suggests that it may be possible to obtain a universal set of curves giving the temperature dependence of ao and al for all absorption coefficients. We attempt to do this as follows: Considering first a 0, we define for any absorption coefficient an effective Elsasser #, a measure of the non-greyness, by considering the limit 62 --* oo. For the Elsasser band this limit is given by equation (5 I), whereas in general equation (49) describes this limit. By equating these two results, we define the effective fl, say freff o) for an arbitrary absorption coefficient as
r [o dvfo dv'B(v)Blv')a2(v) ]
fl{_o~= coth- 1 U. ° dv J'~-~)dv'B(v)B(v a(v)a(~')J"
(67)
Further, we plot tro/trv versus - s 0 e2, rather than v e r s u s e 2 as in Figs. 1-6. Then, for small to first order, and in the limit 6 2 ----- - - e 2 --} o0, the Elsasser curve, with ffe°~given by equation (67), will represent any absorption coefficient. The question is how well will this representation work for intermediate values of e 2. In the case of tra we use a similar scheme, defining Peff°(l)by comparing equations (50) and (52). This gives e 2, c o r r e c t
"(')
coth- ' [J'• dvj':
allaR
dv'B'Iv)B'(v'la2(v)la(v')]"2
(68)
where B' - OBIt3T.In this case we plot versus - $ 1 e2. We test this idea on the continuum absorption coefficient, cr = x - a ( 1 - e - X ) . In this case P'effR(°)= PeffR(1) = 0, and equations (58) and (60) describe the Elsasser band. From equations (55) and (56), the corresponding values of So and sl are So = ~ and s 1 = ~o. The values of So and st for ~r = x - 3 ( 1 - e -x) have previously been given. The results of this test are
610
G.C. POMRAN1NG
shown in Figs. 7 and 8. If our scheme had worked perfectly, the two curves in each figure would have fallen on top of each other. Clearly the results are not good. Part of the problem seems to be that the small e expansion for the continuous absorption coefficient only holds for extremely small values of e, since it was assumed that e/tr is small, and a approaches zero for large values of the frequency. This is, of course, a very severe test for the scheme since the Elsasser band and the 1Iv 3 behavior represent the two extremes of absorption
1.6
1.4
bo
Etsosserband~-~/
1.2
.OOI
0OI
0.1
I.O
-SOE2 FIG. 7. (7 o f o r
negative e2.
16 1'4 Elsosserbond~.~
b-
/3=o
~Continuum I'00"001
001
.SI ~2
FIG. 8. crI for negative ~2.
OI
I'0
Grey radiative transfer
611
coefficients, i.e. pure lines and pure continuum. It should also be noted that one of the approximations of Section 2, namely that the frequency variation of B is slow compared to that of a, is of questionable validity in the continuum case. Nevertheless, because of the failure of this test, we must conclude that in the one group, or grey case, one cannot, by assuring correctness for small and large 62 , represent with reasonable accuracy for all absorption coefficients, the intermediate dependence of the moment opacities on 62 by a universal set of curves, characterized by a single parameter ~,,rf.t~t"~However, one can still hope that in the multigroup problem, in which case the continuum is essentially independent of frequency within a group (if enough groups are used), line effects on multigroup opacities can be reasonably well represented by such a single set of curves. Numerical calculations on real absorption coefficients should be performed to test this conjecture in the multigroup case. 4. C O M P A R I S O N
WITH AN EXACT
MULTIFREQUENCY
RESULT
To assess the accuracy of the grey equation suggested in this paper, we consider a model problem for which the exact multifrequency result is easily obtained. Let a homogeneous halfspace, occupying 0 < z < oo, have an absorption coefficient given by the Elsasser band, equation (42). We wish to compute the flux leaving the halfspace if the temperature in the halfspace is given and no flux is incident at z = 0. This flux is given by F(0) =
dyE(0, v) = 0
-
/ d#gI(0, v,
0
.
(69)
-1
Equation (1) is the relevant transport equation. Since it is a first order equation, it is easily solved for l(z, v, I~).Then, integrating over angle as indicated in equation (69), we find OO
F(O, V)
21t I- dztr(v)B(T, v)E2[a(v)z],
(70)
0
where
E2(~) is the
usual exponential integral of second order 1
E2(~) =
f dge-¢/u
(71)
0
Now, integrating equation (70) over all frequency, interchanging the order of integration, and assuming a(v) is periodic with periodicity d, we obtain
F(0)=
f dzTT'*(Z)-df dva(v)E2[tr(v)z], 0
(72)
0
where y is a constant given by equation (3). We now assume an exponential temperature distribution of the form
~T'*(z) =
e ~:
(e < 1).
(73)
612
G . C . POMRANING
With this temperature distribution the spatial integral in equation (72) can be carried out. The result is
? oF°/.f o/-,].
F(0) = 2 3 dx ~k ~
~Z-~_~!
o
where o = a(x) is the Elsasser band given by cosh/3 + 1
o(x) - cosh/3-cos x"
(75)
Equation (74) has been evaluated numerically for various values of fl, a measure of the nongreyness of the Elsasser band, and e, a measure of the temperature gradient. We now compare this exact result with the usual Planck and Rosseland grey equations, as well as the grey equation proposed in this paper. Specifically, we consider N = 1 in equation (13); thus equations (16) and (17) are the grey equations we shall consider. We first dispose of the Planck and Rosseland grey equations, which are of the form
OI(z, 1~)
/a cqz where c/is either
=
ff[yT'*(z)-
a e, the Planck mean, or ae,
I(z,/~)],
(76)
the Rosseland mean. This leads to the result
The grey equations given by equations (16) and (17) are considerably more difficult to deal with because of the scattering-like terms they contain. Although exact solutions could undoubtedly be obtained using singular eigenfunction methods, we content ourselves with an iterative solution. Specifically, we compute the asymptotic solution valid for large z, use this result in the scattering-like terms, and solve the resulting equations for F(0). For the present purposes, this procedure should give results of sufficient accuracy. We first consider equation (17). The asymptotic solution is easily found to be, with 7 T4(z) = e~Z,
1.sy(z, ~ )
2 e a pe ~z
=
.
(78)
(ax +e")[ 2~+(°°-al)ln(°a +e] -el j Use of this result in the scattering-like term of equation (17) yields the equation of transfer
kt ~+trll(z,
2eaee~z #) = 2g +(aO_al)in(~
).
(79)
The solution of this equation for the flux leaving the halfspace is given by
I
+
+ fl ~a~ - e /
(80)
Grey radiative transfer
613
The treatment of equation (16) leads to similar, but somewhat more complex, results. The asymptotic solution is given by I,,,y(z, ,u) = ap + ,uJ e= ' ao + sp
(81)
where,
j =
~Cro-elJ 2e2+ 3tr°(a°-a')[ 2e-tr°ln[tr°+el]'Woe/J L
(82)
Use of this result in the scattering-like term of equation (17) gives
#~
+ troI(z,/z) =
(trp + J/z)e~z,
(83)
the solution of which yields
F(O) = ---~-Le
e
L~ e i
Xao-el
e
2/"
(84)
In both this result and equation (80), tr0 and ~l depend upon e as given in Figs. l and 3. We now wish to compare our four grey results, namely equation (77) with 6 = tre, equation (77) with ~ = trg, equation (80), and equation (84) with the exact result given by equation (74). Figures 9 and 10 show, for two values of fl, the ratio of these grey results to the exact result as a function of e, a measure of the temperature gradient. Also, if we define the grey error, E, as E = f¢,~t(0)- tgrey(0) F~,,aet(0) '
(85)
1.0
0-9
0,8 L~
0.7
I
d
0'2
0"4
I
~
0.6 e-2
FIG. 9. Halfspaceflux for fl = 0-1.
f
0"8
I0
614
G . C . POMRANING 1.0
"-
,
. (17)
09
LL
08
X(&
=crl:)/exae '(TE~i
07
o!2
o14
o'.6
0'6
,-o
E2
FIG. 10. Halfspace flux for/3 = 1.0.
we find, for small e, E(equations (76) and (77) with ff = a~,) = 321t _ l - ~a- ~e l] ~ +e 0 ( e E(equations (76) and (77) with ff = a , ) = 2If L ( -4~ - 1
2),
~ + o(e3),
(86) (87)
E(equations 16, 84) = 6L(cr2 > ~ r ~ / ~ + O t ~ j,
(88)
E(equations 17, 80) =
(89)
1 ~r~+0(~3),
where ( a 2) is defined as
1 = _1/'dx 1 ( a 2) rc J a2(x) '
(90)
0
with ~r(x) given by equation (75). F r o m these small e results, we see that the usual grey equation employing the Planck mean is clearly least accurate for small e, since it is the only one with an error of order e, the remaining three all containing errors of order e2. F r o m equations (86) through (89), one can easily rank the four grey equations as to accuracy, at least for small e. We have Equation ,, ,, ,,
(76) with 6 = (76) with 6 = (16), i.e., a* = (17), i.e., ~r* =
ap oR ao ~ri
increasing error
This same ranking is borne out by the more general results given in Figs. 9 and 10.
Grey radiative transfer
615
Based on the results from this model problem, we feel confident in suggesting that the grey method proposed in this paper be used in preference to the usual Planck and Rosseland grey equations. Our results also indicate that, in choosing between equations (14) and (15), the choice tr* = trN should be made. In particular, for N = 1, better accuracy is obtained by setting tr* --- trl rather than tr* = tro. This implies the use of equation (17) as the grey equation of transfer, rather than equation (16). Acknowledgements--This work was supported by the United States Government under contract DASAOI70-C-0028 monitored by the Defense Atomic Support Agency.
REFERENCE I. G. C. POMRAN1NG,On the definition and use of grey opacities in radiative transfer problems, Transport Theory and Statistical Physics 1, 25 (1971).
GREY
RADIATIVE
TRANSFER
G. C. POMRANING Science Applications, Inc., P.O. Box 2351, La Jolla, California 92037
Abstract--The general problem of reducing a frequency-dependent radiative transfer problem to an equivalent grey problem is discussed. In particular, we construct, from the integral transport equation, an asymptotic solution to be used in forming grey opacities. This is closelyrelated, but not identical, to the so-calledB-N method widely used in neutron transport work. This leads to generalizations, to non-zero temperature gradients, of the standard Planck and Rosseland mean opacities. One also obtains the mean opacities relevant to higher angular moments of the equation of transfer. We also show how the angular dependence of this asymptotic result can be incorporated into the grey equation of transfer without introducing an angularly dependent opacity. One needs only to modify the scattering kernel. The dominant term in the modification depends upon the differencebetween the Planck and Rosseland mean opacities. Our resulting grey equation is shown to be exact in the two limiting cases of optically thin and thick systems where it is well known that the standard Planck and Rosseland means, respectively, are the appropriate averages. Numerical results are presented for idealized absorption coefficients, such as the Elsasser band (an infinite array of equally spaced, equal strength Lorentz lines) and a 1/v a behavior (characteristic of bound-free and free-free transitions). We calculate the generalized Planck and Rosseland means and exhibit their dependence upon the temperature gradient. Results from our grey equation of transfer are compared with an exact frequency-dependent result for a model problem and shown to be significantly more accurate than those predicted by the usual grey models employing the standard Planck and Rosseland means. All of our considerations apply to the formation of a multigroup equation of transfer as well as to a grey, or one group, equation.
I. I N T R O D U C T I O N IN PRACTICALapplications of the theory of radiative transfer, a c o m m o n simplification is to use a grey, or o n e group, e q u a t i o n of transfer rather t h a n the correct m u l t i f r e q u e n c y equation. This involves the use of a b s o r p t i o n coefficients averaged over frequency. The usual procedure is to e m p l o y a single m e a n a b s o r p t i o n coefficient in both the emission a n d a b s o r p t i o n terms. This m e a n is generally t a k e n as the P l a n c k mean, a n average of the a b s o r p t i o n coefficient over the P l a n c k function, or the R o s s e l a n d mean, a n average of the m e a n free path over the t e m p e r a t u r e derivative of the P l a n c k function. It is well k n o w n that such grey e q u a t i o n s are only correct in limiting cases. The P l a n c k m e a n is a p p r o p r i a t e in the optically thin, emission d o m i n a t e d limit, a n d the R o s s e l a n d m e a n is correct in the optically thick, a l m o s t u n i f o r m t e m p e r a t u r e limit. I n this p a p e r we propose a new grey e q u a t i o n to bridge the gap between these two limiting cases. The analysis involves two distinct steps: (1) the i n t r o d u c t i o n of a n g u l a r effects, which are necessarily present in a grey e q u a t i o n , w i t h o u t i n t r o d u c i n g a n a n g u l a r l y d e p e n d e n t opacity; a n d (2) the calculation of a n a s y m p t o t i c s p e c t r u m to use as a weight f u n c t i o n in f o r m i n g grey opacities. O u r ideas are similar to those found in the n e u t r o n t r a n s p o r t theory literature. The success of this related a p p r o a c h in that field p r o m p t e d the present investigation in the radiative transfer context. 597
598
G . C . POMRANING
We apply the results of our analysis to two idealized absorption coefficients, namely the Elsasser band representing a pure line spectrum, and the 1Iv 3 continuum cross section. We also compare results from our grey equation, as well as results from the usual Planck and Rosseland grey equations, with an exact frequency dependent calculation involving the Elsasser band. The new grey equation is, at least for this model problem, more accurate than the Planck and Rosseland grey equations. It should be pointed out that the analysis given here applies to the development of a multigroup equation as well. In fact, our approximations are more valid in the multigroup case than they are in the grey, or one group, case. We chose to work in this paper with the grey case for simplicity of presentation, as well as the recognition that grey equations of transfer are widely used in practice. 2. R E V I E W O F T H E G R E Y M E T H O D
The details of the development of our grey equation of transfer have recently been given3 t) For completeness in the present paper, we summarize the analysis, omitting much of the algebraic detail. With the neglect of scattering, the time-independent equation of transfer in a planar atmosphere is
#
cgl(z, v, #) - a(z, v)[B(T, v ) - l(z, v,/0], ~z
(1)
where z, v, p, and T are the standard space, frequency, angle (measured with respect to the positive z axis), and temperature variables, I(z, v, I~) is the specific intensity of radiation, a(z, v) is the absorption coefficient suitably corrected for induced emission, and B(T, v) is the Planck function if local thermodynamic equilibrium is assumed. Integration of equation (1) over all frequency yields
#
(~I(z,#) = ap(z)~T'(z)--~(z, 12)l(z,#), dz
(2)
where ~ is a constant given by
7 = 2~4k4/15c2h 3,
(3)
and I(z, iz) is the frequency independent, or grey, specific intensity defined as
l(z, lO = r dvl(z, v, #).
(4)
o
Equation (2) also contains two mean interaction coefficients, namely the Planck mean
ae(z) =
~ dva(z, v)B(T, v) ~o dvB(T, v) '
(5)
and
~ dza(z, v)l(z, v, 12) ~(z,/~) =
~o dvI(z, v,/~)
(6)
Grey radiative transfer
599
The frequency independent equation of transfer, equation (2), is exact. However, to compute 6(z, p) one needs know l(z, v, ~) which, of course, is presumed unknown. Hence, in practice one must replace I(z, v, #) in equation (6) by a weight function, i(z, v, p), which should be an approximation in some sense to l(z, v, #). The entire question of what is an appropriate weight function constitutes the grey opacity problem. We shall return to this point shortly. At present, however, we note that by introducing two grey opacities in equation (2), namely trp and 6, we have assured ourselves of the correct limiting behavior for an optically thin system. In this case, l(z, #) is much less than 7T 4 and hence the choice made for 6 is irrelevant. The only important thing is that the emission coefficient be given by the Planck mean, and this is the case in equation (2) no matter what choice is made for 6. However, we note in general that 6 depends upon angle which detracts from the appeal of equation (2) as a grey equation of transfer. Accordingly, we consider an alternate scheme. As an intermediate step in the development, we expand the specific intensity in Legendre polynomials according to
l(z, v, I-O = . ~= o ~2n + 1 I . ( z ,
(7)
v)P.(#).
Equation (I) is then equivalent to an infinite set of equations for the expansion coefficients
I,(z, v), the nth equation being of the form n
0I._ x(z, v)
Oz
~-(2n+ 1)a(z, v)l.(z, v)+(n+ 1)
,91.+ 1(_7, v)
~3z
= 0.
(8)
If we define the grey moments as
I.(z) = f dvI.(z, v),
(9)
integration of equation (8) over all frequency yields the grey moment equation 01. 1(Z) l-(2n + 1)a.(z)l.(z) + (n + 1) 0_~,)I_(z= O. GZ 0z 1
n - -
(1 O)
Here a.(z) is given by
o.(z) =
5~ dva(z, v)t.(z, v) S~ dvI.(z, v)
(1 I)
As before, since the I,,(z, v) are not known one must use an approximation, say I,(z, v), in equation (1 I) to compute the moment opacities. Now, since equation (7) represents a convergent expansion, the I,,(z, v) and hence the I,,(z) are arbitrarily small for n large enough. Hence, for large n, say n > N, the coefficient of I,,(z) in equation (10), namely a,,(z), is unimportant. That is, as long as a,,(z) is a number of reasonable magnitude, its precise value is irrelevant, for large enough n. In view of this, we replace a,,(z) in equation (10) by tr*(z) for n > N. That is, for n > N we use a common
600
G, C. POMRANING
grey opacity, independent of n. With this replacement, the infinite set of grey moment equations are entirely equivalent to the grey equation of transfer
OI(z, la) ap(Z)TT4(z)- a*(z)l(z, #)+ f d~'ac(z, ~ . ~')l(z, I~'). I~ 8z =
(12)
4n
Here
ae(z, ~ . ~')
is an effective scattering kernel given by
ae(z, , . ,') = f ~ n l [cr*(z)-a.(z)]P,(f~ . ,').
(13)
n:O
In view of the arguments just given, it is evident that tr*(z) is completely arbitrary if N is large enough. In practice, however, N is likely to be chosen as a quite small integer, so the choice of tr*(z) is of some importance. We have considered two choices, namely a*(z) = Cro(Z),
(14)
a*(z) = aN(z).
(15)
and
The supporting arguments for these choices are made in our earlier paper. (1) It is also easily argued that, no matter what choice is made for a*(z), equation (12) is correct in the optically thin limit since the erhission mean is the proper Planck average. As a simple example of equation (12), let us set N = 1. Then, if we choose a* = ao, equation (12) gives OI(z, p) = 0"p(Z)~ T 4 ( z ) -- O'o(z)l(z ,/2)
1
+~ #[ao(Z)-al(z)] f d.'ll'I(z,l~').
(16)
-1
On the other hand, setting a* -- a 1, we find equation (12) yields, with N --- 1, 1
OI(z, I~) _ ap(Z)TT4(z)--al(z)I(z, !a)+½lax(z)--ao(Z)] f d!a'I(z, la'). Oz
(17)
-1
Our numerical results in Section 4 indicate that both equations (16) and (17) are more accurate than the usual Planck and Rosseland grey equations, with equation (17) being more accurate than equation (16). It must also be pointed out that the multifrequency equation given by equation (1) contains no scattering, whereas our grey form of the same equation, namely equation (12) or the special cases equations (16) and (17), contain scattering-like terms. These terms make
Grey radiative transfer
601
our grey equation much more difficult to solve than the usual grey equations. However, had equation (1) contained scattering terms, the additional scattering-like terms introduced in forming the grey equation would only serve to modify the physically meaningful scattering kernel. Hence, in this more general case of radiative transfer with scattering, our grey equation is no more complex in form than the usual, but less accurate, Planck and Rosseland grey equations. To complete the specification of the grey equation of transfer, we must compute the moment opacities tT, needed in equation (13). These quantities are well defined by equation (11) and our only task is to obtain an approximation to the I,(z, v) for use in this equation. To this end, we seek an asymptotic solution of equation (1) which yields a generalization, to non-zero temperature gradients, of the equilibrium diffusion theory result for l(z, v, I~). In terms of optical depth z, defined as, 2
T
= I dz'a(z', v),
(18)
p/
0
the asymptotic solution of equation (1) is
I(z, V,l~) = [l +-~lB(z),
(19)
where the operator D is defined by D = 0/0z.
~
(20)
The operator appearing in the denominator of equation (19) is merely a shorthand notation for a Taylor series expansion about/~D = 0. Equation (19) was derived by writing equation (1) in integral form, expanding B(z') appearing in the integrand in a Taylor series about z' = z, and integrating term by term, ignoring edge effects. Equation (19) can be rewritten as ],/202
I(z, v,#) = ( a - # D )
q
1 -~ l + ~ 2 - J B ( z ) .
(21)
The common equilibrium diffusion theory used in astrophysics corresponds to neglecting kt2D2/(1-/~2D2) as compared to unity in equation (21). Physically this means that the temperature is a sufficiently weak function of optical depth that one can neglect its second and higher derivatives. In the present treatment we assume, for the purposes of dealing with this term, that
D2B(z)
= 0~2B(z).
(22)
That is, we assume, in the vicinity of z, that B(z) can be represented by an arbitrary linear combination of exponentials (0c2> 0) or trigonometric functions (ct2< 0), and hence satisfies a Helmholtz equation. If we further assume that 0~2 is a weak function of space, then use of the successively differentiated results from equation (22) in equation (21) gives I(z,v,#)=
I 12~2~(1 1--/t
j
-#D)B(z).
(23)
602
G.C. POMRANING
The special case 0t = 0 corresponds to equilibrium diffusion theory. Reintroducing the variable z in favor of z, equation (23) can be rewritten as
I(z, v, p) = ~
fi dz 8-T]
v).
(24)
Equation (24) is the asymptotic spectrum we sought, except that we need specify ct, which in general is a function of space and frequency, although the spatial dependence has been assumed weak. From equation (22), assuming fi is a slowly varying function of space (this is necessary, as we shall see, ifa is to be a weak function of z), we find
(l ~B~d2T ll(~2BI[dWI2
(25)
Now, this entire paper has presumed that fi(z, v) is a strong function of v, due to lines, etc., since otherwise the question of how to form a grey equation of transfer is not a significant one. Accordingly, we assume that the frequency variation of a in equation (25) comes from fi, not B. We then write 0"2~ 2 =
constant in frequency
(26)
=/~2,
and the asymptotic spectrum, equation (24), becomes
I(z,v,#)=
1__/2~82/fi 2
B(T,v).
1 f dz
(27)
Finally, we determine e by using equation (26) in equation (22) and integrating over all frequency. The result is 1 ~2T4 /~2 - - T 4
(28)
~z 2 .
If we consider the asymptotic result, equation (27), as the approximate intensity representation 7(z, v, #) to be used to compute the moment opacities, then the corresponding angular moments needed in equation (l 1) are 1
1
72,(z,v)=2rcB(T,v)fd#P2n(#)(l_#~2/fi2 ),
(29)
-1 1
f
i2.+I(Z,V)= --2~ dr dz OB(T,v) or f1 - !
dgP2.+l(~)
(1--~8' )
E/fiE ~"
(3(I)
If we define 1
An(C) =
~
d~Pn(/~) --1
(31)
Grey radiative transfer
603
then equations (11), (29), and (30) yield f~o dYO'(Z,Y)Wn(T , v)hn(,~/~7) ~ dvw.(T, v)A.(e/a) '
a.(z) =
(32)
where w.(T, v)
~ B(T, v) for n even,
(33)
= (OB(T, v)/OT for n odd. In particular, 1
.
Ao(~) : ~ I n / ~ ] '
(34)
and the other A.(¢) follow from the recurrence relation A.+ ~(¢) = / n-~-]-]~ A.(~)- ~
._,(~)---~ 6.o.
(35)
If e is pure imaginary (e2 < 0) then we set e = i6 and define the real quantity A.(rl) = i"A.(irl).
(36)
In this case a.(z) =
~ dvtr(z, v)w.(T, v)A.(6/a) ~ dvw.(T, v)A.(6/a)
(37)
Specifically, we have 1 .4o(r/) = - t a n - ~(r/), t/
(38)
and the recurrence relation -
n
_
,+
1
,39,
for n > 0. Finally, in the limit of small e or 6, we obtain
So dvw.(W, a, = S~ dvw.(T, v)a-" "
(40)
In particular, in this limit a o in the Planck mean and al is the Rosseland mean. Hence ao and al can be considered as generalizations, to non-zero temperature gradients, of the Planck and Rosseland means, respectively. Because of the limiting behavior of tr, to the Rosseland mean, one can easily show that our grey equation of transfer, equation (12), upon making the diffusion approximation, will give the correct equilibrium diffusion theory relationship, i.e. the mean absorption coefficient involved will be the Rosseland mean. As we noted earlier, equation (12) is also correct in the optically thin, emission dominated, case since the emission coefficient is always the Planck mean.
604
G . C . POMRANING 3. A P P L I C A T I O N
TO IDEALIZED
ABSORPTION
COEFFICIENTS
As an example of the use of the asymptotic spectrum to compute the moment opacities, let us consider an infinite array of equally spaced, equal strength, Lorentz lines. This leads to the Elsasser absorption coefficient S
sinh(2nh/d)
a(v) = d [cosh(2nh/d)-cos(2nv/d)]"
(41)
Here S and h are the total area and halfwidth at half maximum, respectively, of a single line, and d is the spacing between lines. We choose a distance scale such that the minimum value of a(v) over v is unity. Then cosh fl + l a(v) = cosh f l - cos(2nv/d)' (42) where fl = 2nh/d. We further assume the spacing d to be small enough so that B and aB/OT are in effect constant in frequency over a distance d. Then, for e = 0, in which case equation (40) is applicable, we find 1)~. dx(cosh f l - c o s x)"- x a.(e = 0) = (cosh fl + J'~ dx(cosh f l - cos x)" " (43) This leads to
ao(e=O)=at,-
cosh fl + 1 sinhfl '
(44)
and a.(e=0)=
[cosh fl + l i p ._ l(coth fl) [ sinhfl ] P.(cothfl) "
(45)
In particular, equations (44) and (45) give the ratio of Planck to Rossetand means as a0(e = 0) = __at,= coth fl, ax(e = 0)
(46)
at
a well known result for the Elsasser band. Figures 1 through 4 show typical results for a o and a 1 for a general temperature gradient (e2 not necessarily small) corresponding to the Elsasser band, as calculated from equations (32) and (37) for n = 0 and 1. These integrals can be carried out analytically, but the resulting expressions are sufficiently complex that we found it more convenient to perform the integrations numerically. These results for ao and al can be considered as generalizations, to non-zero temperature gradients, of the Planck and Rosseland means. The quantities actually plotted in these figures are the ratio of a o to the Planck mean, and the ratio of trl to the Rosseland mean, i.e. ao at,
ao a0(e = 0)
sinh fl cosh fl + 1 ao,
(47)
al
al
aR
ao(e = 0)
cosh fl cosh fl + 1 at.
(48)
and
Grey radiative transfer
605
0-9~:
# b
o
O-S
0
012
0.14
0.16
018
~2
FIG. 1. Elsasser band, ao for positive e2.
It is interesting to note that for e 2 > 0, both ao and ~1 are less than the Planck and Rosseland means, respectively, whereas for ~2 < 0, go and al are greater than gp and trR. This behavior can be shown to be true for any a b s o r p t i o n coefficient. We also note that Ol, the generalized Rosseland mean, is a somewhat stronger function o f ~2 than is o-o.
1"6
1"4 b~
1.2I
1.0I0'01
0"I
li-O
FIG. 2. Elsasser band, cro for negative ~2.
IO-0
606
G . C . POMRANING 1,0
0"95
b=
0"90
b-
02
0.4
0'6
0'8
~2
FIG. 3. Elsasser band, al for positive e 2.
One can obtain relatively simple analytic results in limiting cases. In general, as 32 = - e 2 approaches infinity, equation (37) gives, for n = 0 and 1,
f~ dva2(v)B(T, v) ao = f~ dvtr(v)B(T, v ) '
(49)
f~ dvtr2(v)tgB(T, v)/aT at = f~ dva(v)~B(T, v)/t~T"
(50)
16
1.4
b1.2
~'0 0.01
0.I
1.0 _E 2
FIG. 4. Elsasser band, e I for negative e 2.
I0.0
Grey radiative transfer
607
For the Elsasser band, this leads to O"o
--
' c o t h fl,
(51)
0"1 ~_.g coth2fl.
(52)
tTp
and aR
On the other hand, for small •2 one obtains for the Elsasser band
0"0 = l_soe2+O(e4),
(53)
Gp
and
a-!l = 1 - sle 2 + O(e'*),
(54)
O"R
where
_1F 2 + e -2" -] So = 6L(cos h/~+ 1 ) 2 j ,
(55)
3 Sl - 5(cosh fl+ 1) 2.
(56)
and
In the limit of fl = 0, one can obtain the entire e 2 dependence of these moment opacities in simple analytic form. The results for 0"0 are 0"0 = e ap ~/2[1 - x/(1 -e2)] 1/2'
(57)
or, if 6 2 = _ g 2 ,
ao 0"e
6 4 2 1 4 ( i +62) -
i]i/z
(58)
.
Similarly, for cr~ we obtain 0"1
0[(8 + 3e 2) [1 - x/(1 -- e2)] ,/2 _ 4el1 + ~/(1 - e2)] l/z]
-j.
O"R
(59)
and O"I
1 F4~%/(I '-I- 62)-'~ - 1] I/2 - - ( 8 -
Str - 10 L-
3~2)['4(1 '-l'-62) - 111/2q
~-v~i~i~~+-~-2~]]
i7-2
_]
(60)
As a second example, consider the simple 1/v 3 absorption coefficient, i.e.
a(v) = # { h~To)- a (1--e-h~/kr ).
(61)
608
G . C . POMRANING
The exponential terms in equation (61) constitute the correction for induced emission. We choose a distance scale such that #(To~T) 3 = 1. T h e n o(v) = x - 3(1 - e-X),
(62)
where x = hv/kT, and the generalized Planck and Rosseland means are, from equation (37) with n = 0 and 1,
(x3)
J'~ dxa2(x) ~
t a n - l(6/a)
j'~ dxa(x) ~
tan-l(6/o)
ao =
,
(63)
and 2
I- x'*e" -II-o
_
[- x'ie x -]]-tr
_
.i'~° dx. (x)[~_lL~tan {71 =
s: dx.(x)[
/l
]
'(6/~r)-I
tan '(6/.)-
(64)
1]"
F o r 6 = 0, equations (63) and (64) yield (65)
ao = av = 15/n* = 0-1540, ~4
a~ = aR =
(66)
= 0-005089. 9450 ~
[2+11
.=1 t,, 6
,,7#
F o r small values o f 6 2, equations (53) and (54) give the 62 (or e 2) dependence with s o = 1844 and st = 657,000. Numerical results for other values of 62 are shown in Figs. 5 and 6.
i-6
i'4
IC iO'e
;0-4
fO-5 _E 2
FIG. 5. Continuous absorption, tro for negative e 2
I0"3
Grey radiative transfer
609
1.4
-%
E
i
I-0 10"7
i 0 -a
10 "6
i 0 +$
FIG. 6. Continuous absorption, a I for negativee2. Comparison of these results with those for the Elsasser band, Figs. 2 and 4, shows that the curves are very similar in nature. This suggests that it may be possible to obtain a universal set of curves giving the temperature dependence of ao and al for all absorption coefficients. We attempt to do this as follows: Considering first a 0, we define for any absorption coefficient an effective Elsasser #, a measure of the non-greyness, by considering the limit 62 --* oo. For the Elsasser band this limit is given by equation (5 I), whereas in general equation (49) describes this limit. By equating these two results, we define the effective fl, say freff o) for an arbitrary absorption coefficient as
r [o dvfo dv'B(v)Blv')a2(v) ]
fl{_o~= coth- 1 U. ° dv J'~-~)dv'B(v)B(v a(v)a(~')J"
(67)
Further, we plot tro/trv versus - s 0 e2, rather than v e r s u s e 2 as in Figs. 1-6. Then, for small to first order, and in the limit 6 2 ----- - - e 2 --} o0, the Elsasser curve, with ffe°~given by equation (67), will represent any absorption coefficient. The question is how well will this representation work for intermediate values of e 2. In the case of tra we use a similar scheme, defining Peff°(l)by comparing equations (50) and (52). This gives e 2, c o r r e c t
"(')
coth- ' [J'• dvj':
allaR
dv'B'Iv)B'(v'la2(v)la(v')]"2
(68)
where B' - OBIt3T.In this case we plot versus - $ 1 e2. We test this idea on the continuum absorption coefficient, cr = x - a ( 1 - e - X ) . In this case P'effR(°)= PeffR(1) = 0, and equations (58) and (60) describe the Elsasser band. From equations (55) and (56), the corresponding values of So and sl are So = ~ and s 1 = ~o. The values of So and st for ~r = x - 3 ( 1 - e -x) have previously been given. The results of this test are
610
G.C. POMRAN1NG
shown in Figs. 7 and 8. If our scheme had worked perfectly, the two curves in each figure would have fallen on top of each other. Clearly the results are not good. Part of the problem seems to be that the small e expansion for the continuous absorption coefficient only holds for extremely small values of e, since it was assumed that e/tr is small, and a approaches zero for large values of the frequency. This is, of course, a very severe test for the scheme since the Elsasser band and the 1Iv 3 behavior represent the two extremes of absorption
1.6
1.4
bo
Etsosserband~-~/
1.2
.OOI
0OI
0.1
I.O
-SOE2 FIG. 7. (7 o f o r
negative e2.
16 1'4 Elsosserbond~.~
b-
/3=o
~Continuum I'00"001
001
.SI ~2
FIG. 8. crI for negative ~2.
OI
I'0
Grey radiative transfer
611
coefficients, i.e. pure lines and pure continuum. It should also be noted that one of the approximations of Section 2, namely that the frequency variation of B is slow compared to that of a, is of questionable validity in the continuum case. Nevertheless, because of the failure of this test, we must conclude that in the one group, or grey case, one cannot, by assuring correctness for small and large 62 , represent with reasonable accuracy for all absorption coefficients, the intermediate dependence of the moment opacities on 62 by a universal set of curves, characterized by a single parameter ~,,rf.t~t"~However, one can still hope that in the multigroup problem, in which case the continuum is essentially independent of frequency within a group (if enough groups are used), line effects on multigroup opacities can be reasonably well represented by such a single set of curves. Numerical calculations on real absorption coefficients should be performed to test this conjecture in the multigroup case. 4. C O M P A R I S O N
WITH AN EXACT
MULTIFREQUENCY
RESULT
To assess the accuracy of the grey equation suggested in this paper, we consider a model problem for which the exact multifrequency result is easily obtained. Let a homogeneous halfspace, occupying 0 < z < oo, have an absorption coefficient given by the Elsasser band, equation (42). We wish to compute the flux leaving the halfspace if the temperature in the halfspace is given and no flux is incident at z = 0. This flux is given by F(0) =
dyE(0, v) = 0
-
/ d#gI(0, v,
0
.
(69)
-1
Equation (1) is the relevant transport equation. Since it is a first order equation, it is easily solved for l(z, v, I~).Then, integrating over angle as indicated in equation (69), we find OO
F(O, V)
21t I- dztr(v)B(T, v)E2[a(v)z],
(70)
0
where
E2(~) is the
usual exponential integral of second order 1
E2(~) =
f dge-¢/u
(71)
0
Now, integrating equation (70) over all frequency, interchanging the order of integration, and assuming a(v) is periodic with periodicity d, we obtain
F(0)=
f dzTT'*(Z)-df dva(v)E2[tr(v)z], 0
(72)
0
where y is a constant given by equation (3). We now assume an exponential temperature distribution of the form
~T'*(z) =
e ~:
(e < 1).
(73)
612
G . C . POMRANING
With this temperature distribution the spatial integral in equation (72) can be carried out. The result is
? oF°/.f o/-,].
F(0) = 2 3 dx ~k ~
~Z-~_~!
o
where o = a(x) is the Elsasser band given by cosh/3 + 1
o(x) - cosh/3-cos x"
(75)
Equation (74) has been evaluated numerically for various values of fl, a measure of the nongreyness of the Elsasser band, and e, a measure of the temperature gradient. We now compare this exact result with the usual Planck and Rosseland grey equations, as well as the grey equation proposed in this paper. Specifically, we consider N = 1 in equation (13); thus equations (16) and (17) are the grey equations we shall consider. We first dispose of the Planck and Rosseland grey equations, which are of the form
OI(z, 1~)
/a cqz where c/is either
=
ff[yT'*(z)-
a e, the Planck mean, or ae,
I(z,/~)],
(76)
the Rosseland mean. This leads to the result
The grey equations given by equations (16) and (17) are considerably more difficult to deal with because of the scattering-like terms they contain. Although exact solutions could undoubtedly be obtained using singular eigenfunction methods, we content ourselves with an iterative solution. Specifically, we compute the asymptotic solution valid for large z, use this result in the scattering-like terms, and solve the resulting equations for F(0). For the present purposes, this procedure should give results of sufficient accuracy. We first consider equation (17). The asymptotic solution is easily found to be, with 7 T4(z) = e~Z,
1.sy(z, ~ )
2 e a pe ~z
=
.
(78)
(ax +e")[ 2~+(°°-al)ln(°a +e] -el j Use of this result in the scattering-like term of equation (17) yields the equation of transfer
kt ~+trll(z,
2eaee~z #) = 2g +(aO_al)in(~
).
(79)
The solution of this equation for the flux leaving the halfspace is given by
I
+
+ fl ~a~ - e /
(80)
Grey radiative transfer
613
The treatment of equation (16) leads to similar, but somewhat more complex, results. The asymptotic solution is given by I,,,y(z, ,u) = ap + ,uJ e= ' ao + sp
(81)
where,
j =
~Cro-elJ 2e2+ 3tr°(a°-a')[ 2e-tr°ln[tr°+el]'Woe/J L
(82)
Use of this result in the scattering-like term of equation (17) gives
#~
+ troI(z,/z) =
(trp + J/z)e~z,
(83)
the solution of which yields
F(O) = ---~-Le
e
L~ e i
Xao-el
e
2/"
(84)
In both this result and equation (80), tr0 and ~l depend upon e as given in Figs. l and 3. We now wish to compare our four grey results, namely equation (77) with 6 = tre, equation (77) with ~ = trg, equation (80), and equation (84) with the exact result given by equation (74). Figures 9 and 10 show, for two values of fl, the ratio of these grey results to the exact result as a function of e, a measure of the temperature gradient. Also, if we define the grey error, E, as E = f¢,~t(0)- tgrey(0) F~,,aet(0) '
(85)
1.0
0-9
0,8 L~
0.7
I
d
0'2
0"4
I
~
0.6 e-2
FIG. 9. Halfspaceflux for fl = 0-1.
f
0"8
I0
614
G . C . POMRANING 1.0
"-
,
. (17)
09
LL
08
X(&
=crl:)/exae '(TE~i
07
o!2
o14
o'.6
0'6
,-o
E2
FIG. 10. Halfspace flux for/3 = 1.0.
we find, for small e, E(equations (76) and (77) with ff = a~,) = 321t _ l - ~a- ~e l] ~ +e 0 ( e E(equations (76) and (77) with ff = a , ) = 2If L ( -4~ - 1
2),
~ + o(e3),
(86) (87)
E(equations 16, 84) = 6L(cr2 > ~ r ~ / ~ + O t ~ j,
(88)
E(equations 17, 80) =
(89)
1 ~r~+0(~3),
where ( a 2) is defined as
1 = _1/'dx 1 ( a 2) rc J a2(x) '
(90)
0
with ~r(x) given by equation (75). F r o m these small e results, we see that the usual grey equation employing the Planck mean is clearly least accurate for small e, since it is the only one with an error of order e, the remaining three all containing errors of order e2. F r o m equations (86) through (89), one can easily rank the four grey equations as to accuracy, at least for small e. We have Equation ,, ,, ,,
(76) with 6 = (76) with 6 = (16), i.e., a* = (17), i.e., ~r* =
ap oR ao ~ri
increasing error
This same ranking is borne out by the more general results given in Figs. 9 and 10.
Grey radiative transfer
615
Based on the results from this model problem, we feel confident in suggesting that the grey method proposed in this paper be used in preference to the usual Planck and Rosseland grey equations. Our results also indicate that, in choosing between equations (14) and (15), the choice tr* = trN should be made. In particular, for N = 1, better accuracy is obtained by setting tr* --- trl rather than tr* = tro. This implies the use of equation (17) as the grey equation of transfer, rather than equation (16). Acknowledgements--This work was supported by the United States Government under contract DASAOI70-C-0028 monitored by the Defense Atomic Support Agency.
REFERENCE I. G. C. POMRAN1NG,On the definition and use of grey opacities in radiative transfer problems, Transport Theory and Statistical Physics 1, 25 (1971).