3zpzti NCAR/TN-121 +STR NCAR TECHNICAL NOTE 4 July 1977 The Delta-Eddington Approximation for a Vertically Inhomogeneous Atmosphere W. J. Wiscombe ATMOSPHERIC ANALYSI ANAND PREDICTION DIVISION -~P---P·-·~---~~I----II~-~-YU~ ~eger -- 1~-~ II~- -·IC-- ~ e - -~ - I I I I -- C a NATIONAL CENTER FOR ATMOSPHERIC RESEARCH BOULDER, COLORADO iii ABSTRACT The delta-Eddington approximation of Joseph, Wiscombe, and Weinman (1976) is extended to an atmosphere divided up by internal levels into homogeneous layers. Flux continuity is enforced at each level, leading, as the mathematical essence of the problem, to a penta-diagonal system of linear equations for certain unknown con- stants. Fluxes (up, direct down, diffuse down, and net) are then pre- dicted at each level. Unphysical results of the model are examined in detail. Potential numerical instabilities in the solution are noted and corrected, and an extremely fast, well-documented computer code resulting from this analysis is described and listed. Actual computed fluxes are given for several test problems. v PREFACE It has always been painfully obvious, in my multi-spectral- interval, multi-layer, and multi-angle computer models of solar and IR radiative transfer, that the angular part of the calculation was the pacing item. Since my interest has primarily been in fluxes, it always seemed particularly unfortunate to spend the lion's share of computing time on obtaining angular information, which was then used only to compute fluxes. It would obviously have been preferable to calculate fluxes directly, but none of the existing approximations for so doing (variants of two-stream and Eddington) seemed sufficiently flexible - for in marching through the solar and IR spectrums, and vertically upward through an atmosphere, one encounters huge varia- tions in optical depth and single-scattering albedo, and the existing approximations were only valid for restricted ranges of these para- meters. Even worse, they seemed not to be able to handle the asym- metric phase functions typical of clouds and aerosols very well. Then J. H. Joseph, J. A. Weinman and I (1976) discovered the excellent accuracy of the delta-Eddington approximation (which cal- culates flux directly) for all phase functions, no matter how asym- metric, and for all optical depths and single-scattering albedos; but we investigated only homogeneous layers. It was therefore only natural for me to work up a multi-layer version, which I used to replace adding-doubling in my spectrally-detailed radiation models for a cloudy atmosphere. The solar flux changes from so doing were -5-15 watts/m 25(on the order of present uncertainty in the solar vi constant), and could equally well be caused by almost undetectable variations in the effective radius of the cloud drops. [These results were presented in August, 1976, at the Symposium on Radiation in the Atmosphere in Garmisch-Partenkirchen, West Germany.] I concluded from these results that the multi-level delta-Eddington approximation was potentially of great utility to a wide variety of users who wanted radiative fluxes of roughly 1% accuracy. This was the motivation for publishing the present document. I would like to thank Dr. V, Ramanathan for his careful review of and suggestions for improving this publication. Warren J. Wiscombe May 1977 vii TABLE OF CONTENTS Page 1. ONE-LAYER FORMULAS .......... 0 0 0 0 0 0 0 1 2. ALBEDO OF A SEMI-INFINITE LAYER . 0 *I 0 0 0 0 0 0 a 6 3. MULTI-LAYER FORMULATION . ..... 0 0 0 0 0 0 10 4. THE COMPUTER CODE ........ 0 0 16 4o1 ELIMINATION OF w=1 BRANCH .. ....... 16 4.2 MULTIPLE DIRECTIONS OF INCIDENCE ..... 18 4.3 SOLVING THE LINEAR SYSTEM ...... 18 4.4 ILL-CONDITIONING .. I 20 4.5 SPURIOUS AND NEGATIVE FLUXES IN HIGHLY ABSORBING CASES . .. .. .. 22 4.6 FAILURE TESTING .. ................. 27 4.7 SPEEDING UP THE CODE . ................ 27 5. ACCURACY . 29 REFERENCES .... ... .................... 30 APPENDIX A: LISTING OF COMPUTER CODE FOR THE MULTI-LAYER DELTA-EDDINGTON APPROXIMATION ........ ... 31 APPENDIX B: TEST PROBLEMS 1-11 .... ...... ...... 53 APPENDIX C: COMPARISON OF DELTA-EDDINGTON AND ADDING-DOUBLING FLUX COMPUTATIONS ................. 63 1 1. ONE-LAYER FORMULAS The delta-Eddington formulas for a single homogeneous layer will be required in the multi-layer formalism. Let us therefore derive them, using the notation: w = scaled single scattering albedo of layer g = scaled asymmetry factor of layer T = scaled optical depth within layer AT = total scaled optical depth of layer \P = cosine of zenith angle of monodirectional beam incident upon top of layer S = incident-beam flux A = surface albedo. The scaling relationships are = * 2 fl-e* )0*2 g T = ( w*g**g*2)T* )* , (1 - ) -*- +l~ g - w*g*W*~~g1 where g*, T*, a* are the actual values of the layer's optical parameters. If we follow the procedure outlined in the paragraph following Eq. (16) of Joseph et al. (1976), we arrive at two differential equa- tions, which are: dG 3 3 3ST/g 0 dT + 2 (l-g)H = wSgpe (la) dH 1 -T/P0 d + 2(1-w)G = wSe (lb) where G, H are proportional to the i0, i1 of Joseph et al., E G -E i0 , H -2 ri (2) 2 The solution of these equations is straightforward when w and g do not vary with T. There are two cases which arise: (1) w < 1 G = cle + c2 ae O (3a) X- -XT T/1Oi H = -Pcle + Pc2e / - e (3b) 2 where cl, c2 are arbitrary constants and X = /3(1-o)(l-wg) (3c) I ..... (3d) 2 = X | (3d) p 3 1-cgX , - , 3 1 + g(l-) (3e) 4 2 2 (l/po)-X 3g(l-w)pO0 + (1/p0) =3 WS 2 2 - (3f) 2 2 (1/) X (2) = 1 G = c1 - (l-g)c 2 T - 4 Se (4a) 2 1 S-/ 0 H = c 2 - Spe 0 (4b) where cl, c2 again are arbitrary constants. The solution (3a-f) is invalid when p0 = 1/X, for then the particular and homogeneous solutions of the differential equations (la-b) coincide. Eqs. (3a-b) do approach a finite limit as p0 - 1/X, but computational overflows will occur in practice, since the limit 3 > is of the form 0/0. However, D 0 can only equal 1/% if X 1, or if g+l - - g + 1 w - l = 2g < Since dol/dg 0, and since wo1 equals 2/3 at g = 0, it is clear that we need never concern ourselves with this case if w > 2/3. Even should it arise, it can easily be avoided by changing pO slightly. Only upon attempting an analytic integration over pO might it be necessary to bear this case in mind (numerical integration could simply avoid picking pO = 1/X as a quadrature point). Diffuse up, diffuse down, and net fluxes for the layer are given by the following expressions: -1 -1 F = 27 r pi(T,11) dp = 27 1 (Di0 + i12i1 ) dl 0 0 2 = 7(i 0 - i1 ) = G- H (5a) +, .1 2 F = 27 (5b) I pli(T,v) dp = Tr(i 0 + il) = G + H 0 -T/PO + F + -T/pO F = p Se + F - F = p Se + 2H (5c) Note from Eqs. (5c) and (lb) that d = -Se + dH +/0 ^ -= - - 2 = -(1-c) (Se 0 + 4G 1 (6) This shows that when there is no absorption, w = 1, the approximation 4 conserves flux. Furthermore, when absorption is present, w < 1, the approximation guarantees positive absorptivities, dF/dT < 0, provided G > 0. [Clearly G, being proportional to mean intensity, should be positive on physical grounds.] The differential equation satisfied by G, from Eqs. (la-b), is 2 3 -T/P0 G" - X G = - w [l+g(l-o)]Se Because the right-hand side is negative, G cannot have a relative minimum at which it is negative or zero; for at such a minimum, G" > 0 and X2 G < 0, which is impossible. Hence G(O) > 0 and G(AT) > ==0 G(T) 0 for O < T < AT because otherwise G would, upon leaving G(O) 2 0 and becoming negative, have to 'turn back,' constituting thereby a relative minimum, to reach G(AT) 2 0. One can argue for G(O) 2 0 as follows. The top boundary condi- + tion, prescribing a diffuse down-flux F0 at T = 0, is G(0) + H(O) = F0 . (7) But physically the net flux at the top of the layer cannot exceed the incident flux 10S + F0 , which, from Eq. (5c), implies H(O) < - F . Therefore G(O) > 0 from Eq. (7). [This is not, of course, a rigorous proof, in that it draws on an assumption not properly part of the equations.] Given G(0) > 0, one can argue rigorously for G(AT) > 0. Let + as Lambert there be a prescribed up-flux F 0 at T = AT, as well 5 reflection with albedo A(p ): G(AT) - H(AT) = F0 t + A(D 0 ) G(AT) + H(AT) + p 0Se i (8) [The reason for a p0 dependence in A is explained later.] Eq. (la) allows this to be reduced to a boundary condition involving G alone, G'(AT) + Y1G(AT) = where all we need to know about y1 and y2 is that they are positive. Suppose G(AT) < 0. Then G'(AT) > 0 from the last equation. It follows that G(T) must have a negative relative minimum in order to turn up and reach G(0) 2 0. Since this is impossible, G(AT) > 0 is proved by con- tradiction. In sum, positivity of G(0) establishes positivity of G, and therefore of absorptivities, at all optical depths.