Basics about radiative transfer

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1 aic about radiativ tranfr runo Carli Day Lctur aic about radiativ tranfr - runo Carli

2 Tabl of Contnt Th radiativ tranfr quation. Th radiativ tranfr quation in a impl ca Analytical olution of th intgral quation of radiativ tranfr Th complication of th problm Scattring Non-local thrmodynamic quilibrium Variabl mdium Dicrt calculation of radiativ tranfr intgral Optical path: rfractiv ind of th atmophr and mirag. Th grn flah Scintillation Day Lctur aic about radiativ tranfr - runo Carli

3 Radiativ Tranfr Equation Th pcific intnity of radiation i th nrgy flu pr unit tim, unit frquncy, unit olid angl and unit ara normal to th dirction of propagation. Th radiativ tranfr quation tat that th pcific intnity of radiation during it propagation in a mdium i ubjct to lo du to tinction and to gain du to miion: d d = µ ρ j whr i th co-ordinat along th optical path, µ i th tinction cofficint, ρ i th ma dnity j i th miion cofficint pr unit ma. Day Lctur aic about radiativ tranfr - runo Carli 3

4 A Simpl Ca A a tart it i uful to tudy th radiativ tranfr quation in th impl ca of: no cattring ffct, local thrmodynamic quilibrium, homognou mdium. Day Lctur aic about radiativ tranfr - runo Carli 4

5 Etinction n gnral, th tinction cofficint µ includ both th aborption cofficint and th cattring cofficint, of both th ga and th arool prnt in th ga: ga ga arool µ = arool n th ca of a pur ga atmophr with no-cattring a impl prion i obtaind: ga µ = = Day Lctur aic about radiativ tranfr - runo Carli 5

6 Emiion n abnc of cattring and for local thrmodynamic quilibrium LTE, th ourc function i qual to : ρ j = T whr i th aborption cofficint qual to th miion cofficint for th Kirchhoff' law and T i th Plank function at frquncy and tmpratur T. Day Lctur aic about radiativ tranfr - runo Carli 6

7 Radiativ Tranfr Equation for LTE and No Scattring For an atmophr with no cattring and in LTE th radiativ tranfr quation i rducd to: d d = T Day Lctur aic about radiativ tranfr - runo Carli 7

8 Not on Conrvation of Enrgy d d = T Lo and gain mut oby th cond law of thrmodynamic. For any trm that introduc a lo thr mut b a trm that introduc a gain. n th propagating bam a chang of intnity i caud by th diffrnc btwn th intnity of th ourc that i bing attnuatd and th intnity of th local ourc T. Day Lctur aic about radiativ tranfr - runo Carli 8

9 Analytical Solution of th ntgral Homognou Mdium An analytical intgral prion of th diffrntial quation of radiativ tranfr: d d = can only b obtaind for an homognou mdium. T Day Lctur aic about radiativ tranfr - runo Carli 9

10 Day Lctur aic about radiativ tranfr - runo Carli 0 DRAGON ADVANCED TRANNG COURSE N ATMOSPHERE REMOTE SENSNG Analytical Solution of th ntgral Homognou Mdium Th diffrntial quation i at point and w want to obtain th intgral from and. Thi can b formally obtaind multiplying both trm of th diffrntial quation by p[ ] i.. th attnuation from to. An prion i obtaind that can b intgratd from to. ] [ T d d = T d d = [ ] [ ] T d d d d = T = = T T =

11 Analytical Solution of th ntgral Homognou Mdium n th intgral prion of radiativ tranfr: = T th firt trm i th Lambrt-r law which giv th attnuation of th trnal ourc and th cond trm giv th miion of th local ourc. Day Lctur aic about radiativ tranfr - runo Carli

12 Th Complication Th modlling of radiativ tranfr i mad mor complicatd by : cattring, non-lte, variabl mdium. Th will b individually conidrd, th imultanou application of mor than on complication i only an analytical problm. Day Lctur aic about radiativ tranfr - runo Carli

13 Scattring n prnc of cattring: ga ga th diffrntial quation i qual to: arool arool µ = = d d = T J lo gain Day Lctur aic about radiativ tranfr - runo Carli 3

14 Scattring out Ω Lo For ach path, th amplitud of th cattrd intnity out Ω in ach dirction Ω i maurd by th cattring pha function p Ω: out Ω = p Ω with: p Ω dω = out Ω dω = p Ω dω = Day Lctur aic about radiativ tranfr - runo Carli 4

15 Scattring in Ω Gain Th amplitud of th intnity in Ω cattrd into th bam from ach dirction Ω i maurd by th cattring pha function p Ω: in Th total contribution i qual to: in Ω = p Ω Ω whr th ourc function J i dfind a: Ω dω = p Ω Ω dω = J J df Ω Ω Ω = p d Day Lctur aic about radiativ tranfr - runo Carli 5

16 Day Lctur aic about radiativ tranfr - runo Carli 6 DRAGON ADVANCED TRANNG COURSE N ATMOSPHERE REMOTE SENSNG Analytical Solution of th ntgral Equation Homognou Scattring Mdium Th diffrntial quation i at point and w want to obtain th intgral from and. Thi can b formally obtaind multiplying both trm of th diffrntial quation by p[ ] i.. th attnuation from to. An prion i obtaind that can b intgratd from to. J T d d = J T d d = J d d T d d d d = ] [ ] [ ] [ [ ] [ ] J T = [ ] [ ] J T = [ ] [ ] J T =

17 Day Lctur aic about radiativ tranfr - runo Carli 7 DRAGON ADVANCED TRANNG COURSE N ATMOSPHERE REMOTE SENSNG Scattring Thrfor, in prnc of cattring th diffrntial quation of radiativ tranfr i : and th olution ovr an homognou path from to i qual to: J T d d = J T =

18 Rlativ contribution d d T J = Thrmal radiation Sun Moon Plant Atmophr Sun Earth/atmophr Day Lctur aic about radiativ tranfr - runo Carli 8

19 LTE Radiativ tranfr i an chang of nrgy btwn th radiation fild and th nrgy lvl of molcul and atom which ar dfind by th oltzman tmpratur. W ar in local thrmodynamic quilibrium LTE whn th oltzman tmpratur i in quilibrium with th kintic tmpratur. Of cour LTE do not imply a complt quilibrium that includ th radiation fild. Whn an quilibrium it btwn th radiation fild and th local black-body miion no nrgy chang and no radiativ tranfr occur. Day Lctur aic about radiativ tranfr - runo Carli 9

20 Non-LTE Th oltzman tmpratur i controlld by chmical raction, radiation aborption and thrmal colliion. Whn th colliion ar not frqunt nough th oltzman tmpratur can b diffrnt from th kintic tmpratur and w ar in non-lte condition. Whn in non-lte condition w mut conidr th diffrnt componnt of th mdium and dfin for ach of thm thir individual tmpratur T i and aborption cofficint i. Day Lctur aic about radiativ tranfr - runo Carli 0

21 Day Lctur aic about radiativ tranfr - runo Carli DRAGON ADVANCED TRANNG COURSE N ATMOSPHERE REMOTE SENSNG Non-LTE n th ca of non-lte condition, th diffrntial quation of radiativ tranfr quation i : and th olution ovr a path from to i qual to: i i i i i T d d = = i i i i i i i i i T

22 Non-homognou Mdium Whn th optical and phyical proprti of th mdium ar not contant along th optical path, th aborption cofficint and th local tmpratur T dpnd on th variabl of intgration. n gnral, for a non-homognou mdium th diffrntial quation cannot b analytically intgratd. Day Lctur aic about radiativ tranfr - runo Carli

23 ntgral quation of Radiativ Tranfr non-homognou mdium ntnity of th background ourc Tranmittanc btwn 0 and L Tranmittanc btwn l and L L = 0 τ 0, L τ 0 0, L T τ, L dτ Spctral intnity obrvd at L Aborption trm Emiion trm Optical dpth τ, L ' d' = L Day Lctur aic about radiativ tranfr - runo Carli 3

24 Day Lctur aic about radiativ tranfr - runo Carli 4 DRAGON ADVANCED TRANNG COURSE N ATMOSPHERE REMOTE SENSNG Dicrt calculation of Radiativ Tranfr intgral Dicrt calculation of Radiativ Tranfr intgral = = = = N l N l k k k l l N l l l T l L, 0 Thi intgral i numrically prformd a:

25 Day Lctur aic about radiativ tranfr - runo Carli 5 DRAGON ADVANCED TRANNG COURSE N ATMOSPHERE REMOTE SENSNG Tr of opration for dicrt calculation Tr of opration for dicrt calculation gmntation gmntation Spctral intnity obrvd at L = = = = N l N l k k k l l N l l l T l L, 0 Th optical path i dividd in a t of contiguou gmnt in which th path i traight and th atmophr ha contant proprti gmntation. n ach gmnt th aborption cofficint i alo calculatd. Sgmnt of th optical path = l l l

26 Tr of opration for dicrt calculation aborption cofficint Lin trngth Lin hap Cro ction Aborption cofficint numbr dnity k molc l = l k m= η lin m, l = S m, i, T A m, i, i= m, l Χ m, l l m, i, Tl, pl Volum Miing Ratio Spctral intnity obrvd at L l l l = l L l, T N k k N N 0 l= l= k = l Day Lctur aic about radiativ tranfr - runo Carli 6

27 Optical Path n gnral, radiativ tranfr occur along a traight lin and prrv imag. Howvr, radiation i ubjct to rfraction and th rfractiv ind of a ga i diffrnt from that of vacuum. Th rfractiv ind dpnd on th compoition of th ga, but i in gnral proportional to th ga dnity. Whn th bam cro a gradint of dnity it bnd toward th highr dnity. Day Lctur aic about radiativ tranfr - runo Carli 7

28 Optical Path Limb Viw n an atmophr in hydrotatic quilibrium, th air dnity incra with dcraing altitud. A lin of ight clo to th limb viw i ubjct to a curvatur that i concav toward th Earth. Day Lctur aic about radiativ tranfr - runo Carli 8

29 Optical path U of Snll law for optical ray tracing in th atmophr n r θ = n r in γ in nvariant for phrical gomtry: r i inθi = cont Optical invariant for phrical gomtry: r inθ cont ri n i i = Day Lctur aic about radiativ tranfr - runo Carli 9

30 Optical Path Mirag Mirag ar multipl imag formd by atmophric rfraction. A mirag can only occur blow th atronomical horizon. Svral typ of mirag ar poibl, th main on ar: th infrior mirag, whn a rflction-lik imag appar blow th normal imag th uprior mirag, whn a rflction-lik imag appar abov th normal imag. Day Lctur aic about radiativ tranfr - runo Carli 30

31 Optical Path nfrior Mirag Th infrior mirag occur whn th urfac of th Earth, hatd by th Sun, produc a layr of hot air of lowr dnity jut at th urfac. Grazing ray bnd back up into th dnr air abov: Day Lctur aic about radiativ tranfr - runo Carli 3

32 Optical Path nfrior Mirag Day Lctur aic about radiativ tranfr - runo Carli 3

33 Optical Path Suprior Mirag Th uprior mirag rquir a mor compl atmophric tructur with a cold and high dnity layr at om altitud abov th urfac. Day Lctur aic about radiativ tranfr - runo Carli 33

34 Optical Path Suprior Mirag Day Lctur aic about radiativ tranfr - runo Carli 34

35 Th Grn Flah n 700 vral cintit rportd th obrvation of a grn flah jut aftr unt. Th obrvation wr mad ovr th Tirrhnian a. Nwton had jut dicovrd th complmntarity of colour and gav a quick planation of th grn flah a an optical illuion. Th grn flah wa forgottn for about 50 yar, until th obrvation of th mor rar vnt of a grn flah at unri ovr th Adriatic a. Now bautiful pictur of th grn flah can b found on th wb. Day Lctur aic about radiativ tranfr - runo Carli 35

36 Th Grn Flah Day Lctur aic about radiativ tranfr - runo Carli 36

37 Th Grn Flah Day Lctur aic about radiativ tranfr - runo Carli 37

38 Th Grn Flah Day Lctur aic about radiativ tranfr - runo Carli 38

39 Th Grn Flah At unt and unri th Sun ha an uppr grn rim du to th combind ffct of diffuion and attnuation of blu light. Th grn rim i normally too narrow to b n without optical aid. At th folding point of a mirag, thr i a zon of ky, paralll to th horizon, in which trong vrtical trtching occur. Thi broadn th grn rim into a fatur wid nough to b n. Day Lctur aic about radiativ tranfr - runo Carli 39

40 Scintillation Th rfractiv ind cau not only imag ditortion, but alo intnity variation. Th prnc of a variabl atmophr, bcau of ithr th turbulnc of air or th movmnt of th lin of ight, cau mall movmnt of th lin of ight. Th movmnt introduc a trtching and quzing of th imag, which in th ca of a point ourc alo gnrat intnity variation. Th cintallation ar th intnity variation obrvd a a function of tim for a point ourc. Day Lctur aic about radiativ tranfr - runo Carli 40

Engineering Differential Equations Practice Final Exam Solutions Fall 2011

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