AT622 Section 15 Radiative Transfer Revisited: Two-Stream Models

Transcription

1 AT6 Sectin 5 Radiative Tansfe Revisited: Tw-Steam Mdels The gal f this sectin is t intduce sme elementay cncepts f adiative tansfe that accunts f scatteing, absptin and emissin and intduce simple ways f slving multiple scatteing pblems. We intduce simple mdels t slve the elevant adiative tansfe equatin and demnstate hw they ffe a glimpse at the inticate way in which the adiance field depends n the ppeties f the scatteing and absbing medium. 5. Scatteing as a Suce f Radiatin Phtns flwing alng a given diectin ae emved by single scatteing as Bee's law pedicts. Hweve, these phtns can actually eappea again alng that same diectin when scatteed a multiple numbe f times. In fact, many f the scatteing media f inteest t studies f the atmsphee ae multiple scatteing media, that is media cntaining a sufficient numbe f scattees that phtns tavesing it ae likely t be scatteed me than nce. Multiple scatteing f sunlight, f instance, gives ise t many bsevable phenmena that cannt be explained fm single scatteing aguments alne. F example, single scatteing pedicts a sky that is f unifm bightness and cl cntay t what we bseve. The whiteness and bightness f cluds is als a esult f multiple scatteing. Reflectin f visible and micwave adiatin fm vaius sufaces is lagely influenced by multiple scatteing. Multiple scatteing is thus elevant t many tpics and we need t develp a mathematical desciptin f hw these phtns eappea alng the efeence diectin in de t accunt f it. Cnside a beam f mnchmatic adiatin flwing alng a diectin defined by the vect ξ illuminating a small vlume lcated at and f length ds cntaining scatteing paticles (Fig. 5.). The vlume is taken t be small enugh that nly single scatteed phtns emege fm it. The incemental Fig. 5. Gemety f scatteing f diffuse light. ξ is the unit vect that defines the diectin f the flw and is the vect that specifies the psitin f the vlume element elative t an igin pint. 5-

2 incease in intensity alng the diectin specified by ξ due t the scatteing f this incident beam is, by vitue f the definitin f the phase functin given in Sectin 3, s P (, ξξ, ) s s δ I (, ξ ) = σ scads I (, ξ ) dω( ξ ), (5.) whee σ sca is the vlume scatteing cefficient given ealie and the wavelength dependence n all quantities is undestd. The ttal cntibutin t I(, ξ ) by scatteing f the cmplete diffuse field suunding the vlume is given by the integal f Eqn. (5.), namely which leads t the fllwing definitin s P (, ξξ, ) s s di (, ξ ) = σ scads I (, ξ ) dω( ξ ), (5.) s P (, ξξ, ) s s J (, ξ ) = % ω I (, ξ ) dω( ξ ) such that di (, ξ ) = σ ds J (, ξ ). The quantity J(, ξ ) is the suce f adiatin due t scatteing f diffuse light (smetimes this suce is efeed t as vitual emissin) and ω% is the single scatte albed defined as sca (5.4) σ % ω = σ sca ext and vaies between ze f pue absptin and unity f pue scatteing (the latte cnditin is knwn as cnsevative scatteing) such that the quantity - ω% is the factin f the incident adiatin that is absbed by the small vlume element unde cnsideatin. The mnchmatic adiative tansfe equatin defines the net change in intensity f a beam as it taveses the path element ds thugh a small vlume. The change in intensity as the beam taveses a vlume f atmsphee that bth absbs and scattes {and emits} adiatin is di = di(extinctin) di(scatteing) {di(emissin)}, (5.5) di(, ξ ) =σext[ I(, ξ) J (, ξ)] { di( emisin)}, ds (5.6a) The phase functin is a bidiectinal scatteing functin that is entiely analgus t the bidiectinal functins. 5-

3 afte cllecting the extinctin tem and Eqn. (5.4) f scatteing. It is elevant t nte the similaity f this equatin t Eqn. (4.8a,b) except that the scatteing suce in Eqn. (5.6a) is eplaced by the Planck functin in (4.8b). The adiative tansfe equatin elevant t a hizntally statified atmsphee is di( z, θφ, ) σ sca µ = σext I( z, θφ, ) P( z, θφθφ,, ) I(, θφ ) d σ ( ). 4 4 absb T dz π Ω (5.6b) π whee if it wee nt f the pesence f the integal tem, this equatin wuld be a mee diffeential equatin and the they f multiple scatteing wuld have been wked ut and fgtten lng ag. Example 5.: Vitual Suces? The pblem f sla adiatin multiply scatteed by clud aesl is me cnveniently psed in tems f a suce f cllimated light that entes the clud and by scatteing ceates a vitual suce f diffuse adiatin. This leads t an additinal suce tem in the equatin f tansfe, which may be deived as fllws. Cnside Eqn. (5.6b), and suppse that the glbal intensity I(z, θ, φ) may be expessed as tw cmpnents: I(, z θ, φ) = I (, z θ, φ) I (, z θ, φ) * ne f the diffuse field I * and the secnd I ( z, θ, φ) = I ( z) δ ( θ θ ) δ ( φ φ ) f a cllimated beam f intensity I alng the sla diectin θ = θ, φ = φ. Substitutin leads t and µ d σ sca µ I * I = σext ( I * I ) P ( I * I ) d Ω dz di* σ sca µ = σ ext I* PI* dω I P( θ, φ, θ, φ ) dz di dz = σ exti, whee I ( z) = I ( z ) e σ ( z z)/ µ ext We can als fllw the pcedue f Sectin 4.3 t btain an integal equatin f tansfe that is analgus t Eqn. (4.). Hweve, thee is a fundamental diffeence between this and the equivalent integal equatin that fllws in this way. In Eqn. (4.) the suce functin appeaing in the integand is knwn as a pii (assuming that the tempeatue distibutin alng the path is knwn) and the slutin equies a staightfwad integatin f knwn functins. F scatteing, the suce functin appeaing in the integand unftunately cntains the desied intensity and cannt be evaluated a pii unless sme appximatin is made. The pesence f the intensity in the definitin f J is what cmplicates the pblem f multiple scatteing and why a hst f diffeent appaches exist t vecme it. 5-3

4 5. Multiple Scatteing: A Natual Methd f Slutin A natual slutin t pblems f multiply scatteed light in the atmsphee is ne that decmpses the light field int cmpnents that can be identified with the numbe f times a phtn has been scatteed. This is temed the methd f des f f scatteing and Fig. 5. pvides the geneal gemetic setting f discussing this appach. Suppse light f intensity I entes the medium alng the diectin ξ at the pint whee t (, ξ, ). The amunt f adiatin leaving the distant pint I (, ξ) = I (, ξ) T(,, ξ) is the tansmissin functin defined by the path alngξ alng ξ is (5.7) as shwn in Fig. 5.. We efe t I as the educed unscatteed intensity. When sme light I (, ξ ) at an intemediate pint undeges a scatteing event, a fist de pimay scatteed intensity is geneated at that pint by an amunt defined by Eqn. (5.3). This amunt f intensity pe unit length f path is Fig. 5. Gemety f des f scatteing and gemety f a plane paallel atmsphee f cmputing the pimay scatteed intensity induced by a cllimated suce f sla adiatin f intensity I. (, ) (, ) * ξ σ ext I ξ = J whee, accding t Eqn. (5.3), J may be cnsideed as the suce assciated with the pimay scatteing f I. It theefe fllws that the adiatin fm pimay scatteing f light fm all diectins is P (, ξξ, ) I* (, ξ ) = σ sca I (, ξ ) dω( ξ ). The amunt f this pimay scatteed adiatin that is accumulated alng the path fm is (5.8) I (, ξ) = I (, ξ) T(,, ξ) d. * (5.9) 5-4

5 It is a simple and smewhat intuitive matte t shw that cnstuctin f the intensities assciated with highe de scatteing then fllws fm the epeated applicatin f Eqns. (5.8) and (5.9) such that n P (, ξξ, ) n I* (, ξ ) = σ sca I (, ξ ) dω( ξ ), (5.a) n n I (, ξ) = I (, ξ) T(,, ξ ) d * (5.b) f each intege de n =,, f scatteing. The ttal intensity is theefe the sum f all des f scatteing, namely n n I I I I I I = =, (5.a) n which is cnveniently witten as I = I I * (5.b) whee I* in this case is the ttal diffuse intensity I* = n= I n. An bvius questin t ask is hw many des f scatteing ae equied t appximate the diffuse field t sme given accuacy? The geneal answe t this questin depends n hw many paticles thee ae in the vlume and n hw efficiently the paticles scatte the adiatin. A ugh idea f the effect f the single scatte albed n scatteing is given by the fllwing aguments. Suppse I is an uppe bund n I. Then fm Eqn. (5.8) I* (, ξ ) I σ sca P(, ξξ, ) dω( ξ ) I σ sca = (5.) by vitue f the phase functin enmalizatin cnditin Eqn. (5.9) then becmes = ext I (, ) I* (, ) e σ ξ ξ d whee the expnential fact is the tansmissin functin f the path f length cnditin n I * it fllws that I (, ξ ) I ω ( e ) I % ω. σ ext d d =. Fm the Repeating this pcedue f the next de f scatteing leads t I (, ξ ) I % ω and I n (, ξ ) I % ω (5.3) n 5-5

6 f evey scatteing de n. With the fllwing ntatin it fllws that the diffeence (, ξ) f (, ξ), k ( k) n I I = Iξ I ξ ( k ) (, ) (, ) is n= = I j (, ξ ) I j= k j= k % ω % ω. ( ) j j j % ω % ω I j= % ω I = (5.4) Example 5.: Hw many times des a phtn get scatteed? Cnside the example with ω% =.5 and suppse that we equie I k t diffe fm the actual intensity by an amunt n lage than % f I. It fllws that / I. and that.5..5 j = 7 f the neaest intege value. Thus nly 7 des f scatteing ae equied t mdel the diffuse intensity with a % accuacy when ω% =.5. This simple execise ffes a clea illustatin f the significance f ω% t multiple scatteing. We infe that the numbe f scatteings equied t epesent the ttal intensity deceases as the absptin by the paticle inceases ( as % ω ). F example, many des f scatteing cntibute t the ttal adiatin field in cluds at sla wavelengths whee ω% >.9 but elatively few scatteings cntibute at the infaed wavelengths whee ω% <.5 j 5.3 The Tw-Steam Appximatin On examinatin f the equatin f tansfe, which includes scatteing in eithe its intediffeential its integal fm, ne is cnfnted with the cmplicating pesence f the integal tem that invlves an integatin ve the diectin vaiable. In fact if it ween't f this tem, the equatin f tansfe wuld be but a mee diffeential equatin and the they f multiple scatteing wuld have been wked ut and fgtten lng ag. Thus the essence f the simplificatin that is intduced by the class f simple mdels discussed hee is t appximate, in sme way, the angula shape f the adiance field s as t intduce sme appximatin t this integal tem. T this end thee is a ppety f the adiance field that is utilized t geat benefit by these appximate methds althugh it is nt ften explicitly ealized. This 5-6

7 ppety is illustated in Figs. 5.3a and b in which the zenith adiance distibutin is shwn n descent int the sea (Fig. 5.3a) and deep in a thick clud (Fig. 5.3b). It is appaent that this adiance stuctue appaches sme st f asympttic fm with inceasing depth int the "medium". Eventually sme steady distibutin is eached and all adiances decease at the same expnential ate with inceasing depth ultimately shinking dwn in size but peseving its shape. It is als appaent that this asympttic distibutin can be descibed as sme simple functin f zenith angle (this is the basis f the diffusin appximatins, which we will nt discuss hee). Fig. 5.3 (a) The flux distibutin n a clea sunny day at thee indicated depths in Lake Pend Oielle, Idah (adapted fm Peisendfe, 976). These fluxes ae defined f a cllecting suface inclined at an angle θ as shwn in the inset. (b) Measued intensity as a functin f zenith angle btained fm a scanning adimete n an aicaft as it flew thugh the cente f a deep statifm clud. The lwe cuve is the diffeence between measuement and a simple csine f zenith angle vaiatin (King et al., 99). 5-7

8 While we can appach the develpment f the tw-steam equatins in a numbe f diffeent ways, the end esult is always the same, namely that we aive at equatins f the fm d F t F Q = dt F t F Q (5.5) (a) The Tw-Steam Equatins The Cnceptual Appach The aguments fmulated hee ae simila t thse used in the pineeing wk n adiative tansfe by Schuste in 95. Cnside a paallel, hizntally unifm slab f clud and cnside the fluxes flwing in tw ppsing diectins. 3 We will use the supescipt t efe t quantities assciated with flw in the upwad diectin and a supescipt n quantities elevant t dwnwad flw. The twsteam equatins define the enegy balance f this thin slab f thickness z in exactly the same way as Eqn. (5.6b) descibes an enegy balance f a small vlume f clud. In de t expess the adiative enegy budget f a laye z thick, it is necessay t define the fllwing ptical ppeties: The pptin f the incident flux lst by absptin as the adiatin flws thugh the laye f unit thickness is k abs D ± whee D ± is a measue f the 'diffuseness' f the adiatin field. This paamete me less epesents the mean extensin f the path, elative t the vetical, that a diffuse adiatin field tavels as it penetates the laye. It is a functin f the angula ppeties f the intensity field amng the paametes and epesents ne f the simplificatins mentined abve. If we suppse that the angula distibutin f adiatin that pduces the flux is the same in bth diectins (the magnitudes might be diffeent), then D = D -. Althugh this assumptin is questinable, it tends t be univesally used in tw-steam mdels. The pptinal lss f flux by scatteing is s sca b ± pe unit thickness. Hee we nte that the pcess f absptin is teated diffeently fm scatteing in that a measue f the path length is needed f estimating absptin but this measue is nt needed f scatteing. We will futhe suppse that this scatteing is the same whethe the adiatin flws upwad dwnwad, and thus b = b -. Anthe paamete f elevance is the factin f adiatin f that is scatteed in the fwad diectin. This factin is defined such that f b = (5.6) F a change in flux F defined as psitive upwads, then the change in flux n tansfe thugh the laye z is ± ± m ± F = m( Dk s b) F z± s bf z( ± Q z) (5.7) abs sca sca 3 The elatinship between adiative flux and intensity is expled in the Appendix. The deivatin f the tw-steam equatins given hee fllws the me cnceptual aguments f Schuste. The same equatins can be deived diectly fm Eqn. (5.6b) given sme assumptin abut the intensity field. This altenative deivatin is left f late. 5-8

9 whee the last tem in paentheses epesents intenal suces f F ± in the laye z 4. The fist tw tems n the ight hand side and enclsed by paentheses descibe the lsses f adiatin thugh the pcesses f absptin and scatte, espectively, while the middle tems epesent the incease f flux by backscatte f the ppsing steam. Intducing the definitin f ptical thickness as τ = (k abs s sca ) z whee the minus sign defines τ as inceasing dwnwads fm clud base ppsite t the change in z. On taking the limit z, we btain the tw-flw adiative tansfe equatin ± df ± m ± m =[ D( % ω) % ωbf ] % ωbf ( Q ) (5.8) whee ω% = ssca/(s sca k abs ). All tw-steam methds descibed in the liteatue essentially educe dwn t this equatin. The nly diffeence between the vaius methds lies in hw D, b, and S ± ae specified. One example is t cnside the simple phase functin intduced in Sectin 3.7a, then it fllws that b = ( g)/ whee g is the phase functin asymmety. The adiative tansfe equatin then becmes ± df % ω ± % ω m m = D( ω ) ( g) F ( g) F ( ). % Q ± (5.9) The geneal slutin t Eqn. (5.9) f given suces can be cmplicated. Hee we neglect this tem and cnside nly sla adiatin incident n clud tp assuming this incident flux is puely diffuse (as ppsed t the me ealistic case f a puely cllimated incident flux). While the details f the slutins descibed belw change with the additin f the suce tem f sla adiatin, ntably by intducing a sla zenith angle dependence t the slutins, the gss elatinships between the ptical ppeties f cluds (τ*, ω%, and g) and the diffuse eflectance and tansmittance des nt change. 4 Tw main suces f flux ae usually cnsideed in these mdels. One is the suce f adiatin due t themal emissin, which accding t Kichffs law takes the fm Q ± = k abs πb(t) f emitting clud paticles f tempeatue T. The secnd is the suce f diffuse adiatin that esults fm the single scatteing f a cllimated flux F f sunlight. This suce has the fm ± τ / µ b Q = F e ssca f whee f and b ae the fwad and backwad scatteing factins f the incident flux F and these factins ae functins f the csine f the sla zenith angle µ 5-9

10 Example 5.3: Slutin f suceless atmsphee, pue scatteing Cnside the example f a single laye f 'clud' with S =. F pue scatteing, ω% =, k abs = ± F = m m m ( % τ ) whee m and m - ae cnstants detemined by bunday cnditins and * % τ = ( g) τ is the ptical depth f the entie slab, τ*, scaled by the fact ( - g). The elevance f this scaled paamete becmes appaent by cnsideing an islated scatteing and absbing laye illuminated fm abve by flux F velying a dak suface. Unde these cnditins, the albed f the clud laye is and the tansmittance F () ~ τ R = = ~ (5.a) τ F F ( τ * ) T = = R = F ~ τ (5.b) This esult implies that tw nn-absbing clud layes with diffeent ptical thicknesses τ* and g eflect the same amunt f adiatin when the espective values f τ% ae the same. This is efeed t as a similaity cnditin and implies that it is nt pssible t infe τ* fm a single eflectin a single tansmissin measuement withut infmatin abut g. One f the pblems assciated with the emte sensing f ice cystal cluds is that g is neithe well knwn n well undestd in hw it vaies with diffeent cystal habits. This paamete is well knwn f wate dplet cluds and is quasi-cnstant with a typical value in the ange

11 Example 5.4: Slutin f suceless atmsphee, nncnsevative scatteing F this case, ω% <, k abs > : The slutin t Eqn. (5.9) f a suceless, unifm medium has the fm whee and F ± (τ ) kt = m γ ± e mγ m e kt k = {( % ω ) D[( % ω ) D % ω b]} γ ± = ± ( % ω ) D/ k / (5.a) (5.b) (5.c) whee as abve the cefficients m± ae detemined fm apppiate bunday cnditins. Cnside the same cnditins applied t Eqns. (5.a,b) F (τ*) = F - () = F f an islated laye f ptical thickness τ*. With sme manipulatin f Eqn. (5.a), the albed and tansmittance f the laye can be witten as R = γ γ [e -kτ* e -kτ* ]/ (τ*) (5.a) whee T = γ γ τ * ( )/ ( ) * * * kτ kτ e e ( τ ) = γ γ (5.b) (5.c) As τ*, R R = γ /γ and this is efeed t as the albed f a semi-infinite clud. This epesents the uppe limit t the albed f a clud and since T = and A = R, this is als the uppe limit t the absptin A within the clud. These uppe limits ae detemined entiely by the ptical ppeties ω%, k, g, and D f the clud. Fm the substitutin f Eqn. (5.b) in Eqn. (5.c) tgethe with the definitin f R, it fllws that whee R / [ / s ] = s / [ / ] (5.3) % ω s = % ωg / is anthe similaity paamete (Fig. 5.4b). Equatin (5.3) states that the eflectin by tw diffeent ptically thick cluds ae equivalent when the similaity paamete s is equivalent. 5-

12 Fig. 5.4 (a) The spectal eflectance fm mdeled cluds as a functin f thei paticle size. (b) The similaity paamete as a functin f wavelength f diffeent assumed values f the clud dplet effective adius e. 5-

13 Example 5.5: Pllutin Susceptible Cluds The effect f ship stack effluents n clud ptical depth and clud albed is a tpic f intensive inteest. The simple tw-steam mdel intduced peviusly nw seves t emphasize hw the scaled ptical depth is the diect cntlling paamete n the albed f cluds. We can deduce that ptical depth f cluds is τ πn h * f a clud f depth h cmpsed f N paticles f a size that exceeds the wavelength f adiatin. Inceased wate cntent (ccuing lagely as an incease in ), f instance, can incease the ptical depth f cluds. An incease in numbe cncentatin N can als incease τ* and the sensitivity f ptical thickness τ* t N, f cnstant liquid-wate cntent, is given by * τ * τ N = 3 N. F clud dplets unde sla illuminatin, g is quasi-cnstant and.85. Using this value in Eqn. (5.a), ne btains the fllwing simple appximate expessin τ R. 3 τ f the albed f a clud. We can eadily deive the sensitivity f R t dplet numbe N fm this elatin and expess it in tems f N and R. The esult f fixed liquid wate cntent w is dr R( R) =. dn 3N w Thus, f a given N, the mst susceptible cluds ae thse with R /, but the maximum f R is athe flat - f R = /4 3/4, dr/dn is still thee-fuths f its maximum value. F fixed R, (dr/dn ) w is invesely elated t N, which in the eal pesent atmsphee, can vay by me than tw des f magnitude. The susceptibility drldn (gaphed in Fig. 5.5) eveals a cnsideable sensitivity f clean cnditins e.g., in ceanic and emte aeas (whee N is lw). Thee (dr/dn ) w is seen t appach % (pe cm -3 ); that value wuld mean a eflectance change f. f a cncentatin change f just cm -3. Fig. 5.5 Susceptibility f diffeent cnditins f N and R. (b) The Tw-Steam Equatins Analytic Appach: Eddingtn's Appximatin as an Example 5-3

14 Suppse we cnside azimuthally aveaged quantities I, F = π I µ dµ then I % ω µ d = I p( µ, µ ) I( µ ) dµ If we assume that I I I µ which esembles the fm used in u diffusin appximatin, then I F = π Iµ dµ = π I 3 I F = π Iµ dµ = π I 3 and Fnet = F F = π Iµ dµ = I π 3 3 I ( = F F ) F F = π I I F F = π The secnd appximatin we intduce is the fllwing P( µµ, ) = 3gµµ f the phase functin expansin. If we cnside u adiative tansfe equatin and integate ve each espective hemisphee, then we btain the fllwing tw equatins (igning suces f the mment): di % ω π µ ( µ ) dµ = π I( { µ ) dµ p( µ, µ ) I( µ ) dµ dµ B A di % ω ( ) ( π µ µ dµ = π I { µ ) dµ p( µ, µ ) I( µ ) dµ dµ B A C C 5-4

15 Cnside the fist equatin: di d d Tem A= π µ dµ = π µ [ I Iµ ] dµ = F d Tem A = F Tem B= π ( I Iµ ) dµ = ( I I)π Tem B = π ( I Iµ ) dµ = ( I I)π C % ω Tem = dµ ( I Iµ ) 3 gµ ( Iµ Iµ ) d π µ % ω 3 = dµ [ I µ Iµ gµ Iµ gµ Iµ ] % ω = dµ [I gµ I] CC ( ) ( ) Tem = % ω dµ ( I gµ I) π = % ω[ I gi]( % ω[ gi I]) Afte cllecting tems, we btain 3 df = π I I πω% I gi 3 3 = F F ( F F ) % ω F F g( F F ) % ω % ω = F (4 3 g) F F (4 3 g) F Similaly df 7 % ω % ω = (4 3 g) F (4 3 g) F df = π I I πω gi I = % 3 3 = F F ( F F ) F F g( F F 4 % ω ) 4 7 % ω % ω = F (4 3 g) F F (4 3 g) F

16 whee we can eadily identify df 7 % ω % ω = (4 3 g) F (4 3 g) F % ω t = (4 3 g) 4 4 % ω = (4 3 g) 4 (5.4) (c) Delta-Tw Steam Mdels We have aleady emaked n the scaling assciated with phase functins f the fm p( µµ, ) = fδ ( µ µ ) ( f)( 3 gµµ ) which educes t the fmal tw-steam slutins when g f g = f τ = ( % ω f ) τ ( f ) % ω % ω = f % ω ae used diectly in the slutins. It is usual t emply the secnd mment f the expansin, namely f the scaling fact. 5.4 Geneal Slutins f = χ/5 = g The tw-steam mdel and its geneal slutin ae biefly intduced hee. We will cnside tw kinds f suce functins t epesent thse descibed in ftnte. In develping these slutins, it is useful t intduce tw-steam equatins (Eqn. (5.5)) as fllws whee F is a flux vect L F F = Q F F = F (5.5a) f upwelling (F ) and dwnwelling (F - ) flux at level z'. The dependence f each fact in Eqn. (5.5a) n z' is taken t be undestd. The suce functin vect, 5-6

17 Q Q = Q t depends n z'. The tw-steam tanspt peat is L F d t dz = σ ext t (5.5b) whee we nte the steaming tem is defined elative t z athe than τ as in Eqn. (5.5), which means that the cefficients t and diffe fm thse f Eqn. (5.4) nly by a fact f σ ext. Althugh these define the flux equatins, the fm f this equatin is geneic in the sense that they als equally apply t adiance and the 'n steam' pblem (e.g., Flatau and Stephens, 988). The diffeent fms f the equatin cefficients in Eqn. (5.5), namely t and, define diffeent vesin f a tw-steam mdel. The x matix f cefficients defines the attenuatin matix A t = σ ext t (5.6) and the 'slutin' t the suceless equatin (i.e., Q = ) can be expessed in tems f the matix expnential ( z y) M (, zy) = e A (5.7) whee M is a mapping functin. By vitue f the blck stuctue f A, this mapping matix has a simila fm m (, z y) m (, z y) M(, zy) = m (, z y) m(, z y) (5.8) F the x matix A f the tw-steam equatins, Eqn. (5.7) fllws as κ( zy) κ( zy) e e m (, z y) = f f κ( zy) κ( zy) e e m(, z y) = κ 5-7

18 κ( zy) κ( zy) e e m (, z y) = κ κ( zy) κ( zy) e e m(, z y) = f f (5.9) whee f f = ( t/ κ ) = ( t/ κ ) (a) The Inteactin Pinciple Cnside the tw types f adiative tansfe pblem as psed in Fig The gal f the fist is t deduce the fluxes at the uppe bunday f an islated laye at ZT in tems f the fluxes at the lwe bunday z. Stated, this way the adiative tansfe pblem is an initial value pblem. Its slutin is as fllws. Fist cnside the suceless equatins f which the slutin (assuming cnstant cefficients) is F m (, z y) m(, z y) F ( z) = ( ) y F m (, z y) m(, z y) F (5.3) Unftunately, mst pblems f adiative tansfe ae psed as fllws. Given fluxes incident n the bundaies, what ae the emegent fluxes (Fig. 5.6). These ae tw pint bunday value pblems, which can be slved thugh eaangement f Eqn. (5.3). The elatinship between fluxes ut in tems f fluxes in (and intenal suces) is efeed t as the inteactin pinciple. In eaanging Eqn. (5.3) in the fm f the inteactin pinciple, we btain the elatin between the mapping functins abve and the me classical ppeties f eflectin and tansmissin. Fig. 5.6 Tw types f tansfe pblems, the initial value pblem at left and the me taditinal twpint bunday value pblem defining the inteactin pinciple. 5-8

19 Simple eganizatin f Eqn. (5.3) in its inteactin fm gives the desied emegent fluxes F (z T ) and F - (z) in tems f input fluxes F ( z ) T / m m / m F ( z) =. F ( z) m / m m m m / m F ( zt ) (5.3) whee the ntatin indicating the mapping facts ae defined f the laye (z T,z) is dpped f cnvenience. This identifies the laye diffuse eflectin and tansmissin functins as which ae thse f Eqn. (5.). (b) Adding Suces - Geneal Slutin m( zt, z) Rz ( T, z) = (5.3a) m ( z, z) T T( zt, z) = (5.3b) m ( z, z) We pceed with Eqn. (5.5b) in Eqn. (5.5a) and multiplying bth sides by the expnential f the matix T Az df Az Az e e AF( z ) = e Q( z ) (5.33) dz whee we assume that the attenuatin matix (i.e., the ptical ppeties and t) is independent f z'. Integatin f Eqn. (5.33) fm (z z T ) yields F z = e F z S z z (5.34) ( ) ( ) Az t z ( T) ( T, ) by vitue f the ppety f the matix expnential [ e ] Az = e Az and whee the vect S z z = e Q z dz zt A( zt z) ( T, ) ( ) z (5.35) These esemble the me taditinal integal fm f the adiative tansfe equatin (Sectins 4 and ). Hweve, it cntains the desied emegent fluxes (i.e., the slutin) n bth sides f the equatin as seen me clealy in the expanded fm F ( z) m ( zt, z) m( zt, z) F ( zt) S ( zt, z) =. F ( z) m ( zt, z) m( zt, z) F ( zt) S ( z T, z (5.36) ) 5-9

20 A special slutin aises f pblems in which the medium is illuminated with ze incident fluxes (knwn as vacuum bunday cnditins). Then we btain F ( z ) (, )/ (, ) T SzT z m zt z = (5.37) F ( z) m ( zt, z) S ( zt, z)/ m ( zt, z) S ( zt, z) which ae the paticula slutins t Eqn. (5.5a) f geneal slutins. 5-