Methods for solving the radiative transfer equation with multiple scattering. Part 3: Exact methods: Discrete-ordinate and Adding.

Transcription

1 Lecture. Methos for sovg the raatve trasfer equato wth utpe scatterg. art 3: Exact ethos: screte-orate a Ag. Obectves:. screte-orate etho for the case of sotropc scatterg.. Geerazato of the screte-orate etho for hoogeeous atosphere. 3. uerca peetato of the screte-orate etho: O 4. rcpes of varace. 5. Ag etho. equre reag: L: cchazz... Yag et a. BA: A research a teachg software too for ae-parae raatve trasfer the earth's atosphere. Buet of the Aerca Meteoroogca ocety Avace reag: hoas G.E. a K. taes aatve trasfer the atosphere a ocea Chapter Lu Q. a. Weg Avace oubg Ag Metho for aatve rasfer aetary Atospheres. Joura of Atospherc ceces taes K.. say W. Wscobe a K. Jayaweera uercay stabe agorth for screteorate-etho raatve trasfer utpe scatterg a ettg ayere ea." App. Opt screte-orate etho for the case of sotropc scatterg. OE: A screte-orate etho has bee eveope by Charasekhar the 95s Charasekhar. aatve trasfer 96 over ubcatos. eca the raatve trasfer equato Lecture 8 for azuthay epeet ffuse testy: ' ' ' exp 4

2 or sotropc scatterg the scatterg phase fucto s. Hece we have ' ' exp 4 Let s appy the Gaussa quaratures to repace the tegra Eq.[.] [.] a exp [.] 4 hoogeeous part where - ters a a are the Gaussa weghts costats a are quarature ages or pots. Eq.[.] s a syste of hoogeeous ffereta equatos: outo of Eq.[.] geera souto partcuar souto where the geera souto s a souto of the hoogeeous part of the Eq.[.] eotg the geera souto of Eq.[.] ca be fou as g exp k [.3] sertg Eq.[.3] to Eq.[.] we obta g k a g [.4] We ca f g the for g L k where L s a costat to be etere. ubsttutg ths expresso for g Eq.[.4] we have a k Eq.[.5] gves soutos for -k. a k [.5] hus geera souto s where L are costats. L exp k [.6] k

3 he partcuar souto ca be fou as h exp [.7] 4 where h are costats. sertg Eq.[.7] to Eq.[.] we have h a h [.8] ro Eq.[.8] h s fou as h γ where γ s etere fro γ { } [.9] a Ag the geera souto Eq.[.6] a the partcuar souto Eq.[.7] we have L γ exp k exp [.] k 4 where L are costats to be etere fro the bouary cotos. H-fucto has bee trouce by Charasekhar as H... k [.] Expressg γ the H-fucto Eq.[.] becoes L H H exp k exp [.] k 4 Eq.[.] gves a spe souto for the se-fte sotropc atosphere see L:6.. H H 4 [.3] 3

4 4. Geerazato of the screte-orate etho for a hoogeeous atosphere. Let s coser the atosphere wth o-sotropc scatterg. We ca expa the ffuse testy the cose seres cos ϕ ϕ ϕ o we ee to sove exp 4 4 ϖ ϖ δ o he geera souto ay be wrtte as k L exp φ φ k L are coeffcets to be etere. he partcuar souto ay be wrtte as exp p Z Where Z s the foowg fucto 4 H H Z ζ ϖ he copete souto of the raatve trasfer s Z k L exp exp φ [.4] - Let s geeraze the copete souto Eq.[.4] of the raatve trasfer for the hoogeeous atosphere. he atosphere ca be ve to the hoogeeous

5 ayers each s characterze by a sge scatterg abeo phase fucto a optca epth. OE: f a atospherc ayer has gases aerosos aor cous oe ees to cacuate the effectve optca propertes of ths ayer. or -th ayer we ca wrte the souto usg Eq.[.4]. o spfy otatos et s coser the azutha epeet case.e. so we have L φ exp k Z exp [.5] ow we ee to atch the bouary a cotuty cotos betwee ayers. At the top of the atosphere OA: o owwar ffuse testy [.6] At the ayer s bouary: upwar a owwar testes ust be cotuous [.7] At the botto of the atosphere assug the Laerta surface: rsur [ exp ] Eqs.[.6]-[.8] prove ecessary equatos to f the ukow coeffcets. [.8] 3. uerca peetato of the screte-orate etho: O O s a OA uerca coe base o the screte-orate etho eveope by taes Wscobe et a. O s opey avaabe a has a goo user-gue. oe features: O appes to the hoogeeous othothera pae-parae atosphere. A user ay set-up ay ubers of the pae-parae ayers. 3 Each ayer ust be characterze by the effectve optca epth sge scatterg abeo a asyetry paraeter f the Heyey-Greeste phase fucto s use. 4 A user ay use ay phase fucto by provg the Legere poyoa expaso coeffcets. 5

6 5 A user seects a uber of streas keepg that the coputato te vares as 3. 6 A key probe s to obta a souto for fuxes for strogy forwar-peake scatterg. 7 O aows prectg the testy as a fucto of the recto a posto at ay pot the atosphere.e. ot oy at the bouares of the ayers. O s corporate to the BA raatve trasfer coe. BA ata Barbara O Atospherc aatve rasfer s a OA coputer coe: see troucto to BA 4. rcpes of varace. eca the eftos of refecto a trassso of a ayer trouce Lecture 9. f the soar fux s cet o a ayer of optca epth : ϕ ϕ ϕ Or the geera case: ϕ ϕ * ϕ t r ϕ ϕ ϕ ϕ ϕ c * ϕ ϕ ϕ ϕ ϕ c he prcpe of varace for the se-fte atosphere Abartzua 94: the ffuse refecte testy caot be chage f a ayer of fte optca epth havg the sae optca propertes as those of the orga ayer s ae see L:

7 he prcpes of varace for a fte atosphere Charasekhar 95: he refecte upwar testy at ay gve optca epth resuts fro the refecto of a the atteuate soar fux exp a b the owwar ffuse testy at the eve : exp * [.9] ' * * he ffusey trastte owwar testy at the eve resuts fro a the trassso of cet soar fux a b the refecto of the upwar ffuse testy above the eve : [.] ' 7

8 8 3 he refecte upwar testy at the top of the fte atosphere s equvaet to a the refecto of soar fux pus b the rect a ffuse trassso of the upwar ffuse testy above the eve : exp [.] 4 he ffusey trastte owwar testy at the botto of the fte atosphere s equvaet to a the trassso of the atteuate soar fux at the eve pus b the rect a ffuse trassso of the owwar ffuse testy at the eve fro above: * exp * * exp * [.] ' ' o * * ' '

9 5. Ag etho. Ag etho s a exact etho for sovg the raatve trasfer equato wth utpe scatterg. t uses geoetrca ray-tracg approach a the refecto a trassso of each vua atospherc ayer. trategy: kowg the refecto a trassso of two vua ayers the refecto a trassso of the cobe ayer ay be obtae by cacuatg the successve refectos a trasssos betwee these two ayers. OE: f optca epths of these two ayers are equae ths etho s referre to as the oubg-ag etho. Coser two ayers wth refecto a a tota rect pus ffuse trassso a fuctos respectvey. Let s eote the cobe refecto a tota trassso fuctos by a a cobe refecto a tota trassso fuctos betwee ayers a by a respectvey. 9

10 he cobe refecto fucto s...] [... [.3] OE: Eq.[.3] we use that x x he cobe tota trassso fucto s...] [... [.4] he cobe refecto fucto betwee ayers a :...] [... [.5] he cobe tota trassso fucto betwee ayers a :...] [... [.6] ro Eqs.[.3]-[.6] we f that ; ; [.7] Let s trouce sg that exp fro Eqs.[.6]-[.7] we f exp exp exp exp [.8]

11 exp exp exp exp exp δ [.9] hus we ca wrte a syste of teratve equatos for the coputato of ffuse trassso a refecto for the two ayers the for: Q Q Q exp exp exp exp exp [.3] OE: Eq.[.3] the prouct of two fuctos pes tegrato over the approprate age so that a utpe-scatterg cotrbutos are cue. or stace uerca proceure of the ag etho: As the startg pot oe ay cacuate the refecto a trassso fuctos of a ta ayer of very sa optca epth e.g. -8 that the sge scatterg approxato s appcabe. he usg Eq.[.3] oe coputes the refecto a trassso fuctos of the ayer of. 3 sg Eq.[.3] oe repeats the cacuatos ag the ayers ut a esrabe optca epth s acheve.