12.815, Atmospheric Radiation Dr. Robert A. McClatchey and Prof. Ronald Prinn ( ) ν da Area d Solid Angle d frequency int erval dt time

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1 .85, Atmoheric Rdition Dr. Robert A. McCltchey nd Prof. Ronld Prinn. Eqution of Rditive Trnfer Secific Intenity of Rdition Trnfer I θφ,,z = E ( Energy Ω da Are d Solid Angle d frequency int ervl dt time In n intervl d, we loe intenity by extinction (cttering nd bortion nd gin it by emiion nd cttering. Lmbert Lw: The extinction roce i liner, indeendently in the intenity of rdition nd in the mount of mtter, rovided tht the hyicl tte (i.e. temerture, reure, comoition i held contnt. From Lmbert Lw, the chnge of intenity long th d i roortionl to the mount of mtter in the th nd to the intenity of rdition: extinction loe =α I d where α = volume extinction coefficient ( The rgument tht the extinction roce i liner in the mount of mtter lie with equl force to the emiion roce. Therefore, we write: gin =α J d ( where we hve defined the ource function, J. The extinction coefficient cn be exreed the um of n bortion coefficient ( k nd cttering coefficient ( σ. α = k σ (3.85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge of 5

2 The mot generl roblem in tmoheric rdition, therefore, h ource function coniting of two rt, α J = k J therml σ J cttering (4 where k J ( therml ε =ε = for iotroic emiion. 4π σ J ( ct, on the other hnd, i given by two term, decribing the diffuely cttered rdition nd the ingly cttered incident bem of rdition (the un. d ( θφ,,z =α ( I (,,z θφ ε ( θφ,,z ' ' dω σ P ( θ, φ, θ,' φ' I( θ, φ, z (5 4 π πf σ ex ( α z co θ P ( θ, φ, θ, φ 4π where P ( θφθ,,,' φ' i the cttering he function (or cttering digrm nd i normlized uch tht 4 dω P = where dω i n element of olid ngle. The he of the he function cn π 4 π be uefully chrcterized by ingle number, dω < co η > = (co η where η i the cttering 4 π ngle nd <co η > i clled the ymmetry rmeter (which vrie between nd - nd i for iotoic cttering. Dividing by α ( z, we hve: ε ( θφ,,z =I ( θφ,,z α z d α z ( z 4π ( z σ ' ' P ( θφθ,,,' φ' I( θφ,, z in θ'd θ'd φ' 4πα z σ ( ( α θ θφθ φ α z πf ex z z co P,,, z 4π (6 Let u now introduce the following definition: dz d = α dz d = = co θ (7 where = verticl oticl deth meured from the to of the tmohere = tz=. Note tht thi coordinte differ for ech..85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge of 5

3 We then obtin the following Rditive Trnfer Eqution in differentil form in lne rllel tmohere: (,, φ ε ( σ ( ' ' ' ' ' ' = I (, φ, ( θ φ θ φ ( θ φ θ θ Φ α πα P,,, I,, in d d d 4 ( σ πf ex z co P,,, ( ( α θ θφθ φ 4πα ( φ or,, = I,, φ J,, φ d where we hve defined the ource function, J (,, φ : (8 ε ω ' ' ' ' ' ' J (,, φ = P (, φ,, φ I (,, φ,, φ ddφ α 4π ω π F ex ( P (,,, φ φ 4π (9 nd σ α = ω = Single Scttering Albedo. Let u lo be reminded tht the tmohere in generl contin both ge nd rticulte (erool. Ech h cttering nd bortion roertie tht we need to conider. Thu we hve: α = k ge k erool σ ge σ erool There re mny ce where the hyicl itution enble the neglect of one or more of thee term nd relted imlifiction of the Rditive Trnfer Eqution. We will now exmine few uch ce. Let u firt ly Kirchoff Lw to our Rditive Trnfer Eqution. The rtio of emiion nd frctionl bortion in ny direction of lb of ny thickne in thermodynmic equilibrium equl the blck body intenity. So in non-cttering tmohere, we hve ε ( θ, φ, ε ( 4π = = B ( = Blck Body Function α α B ( T 3 h = c ( e h kt B ( T hc = λ ( e λ 5 hc λkt where ( B i iotroic.85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge 3 of 5

4 Thi give u: ω ' ' ' ' 4π J, φ, = B P φ,,, φ I φ,,, φ, ddφ ωπ F ex P,,, 4π ( φ φ (9 Alo if ll relevnt roertie ( α ω ( zimuth ngle, φ i.e. I (,, φ = I (, nd we hve: (,,B re horizontlly invrint, then I i indeendent of, = I (, J( ( d Ce I: Let u firt conider the imlet ce chrcterized follow: We oberve unlight of viible wvelength through non-cttering tmohere: πf B 4π ( << nd lo B ( << I (, or b Bright, rtificil ource of rdition hining through n tmoheric th (e.g.- ler: We then hve tht ( >> J ( Eqution i imlified to: ( I, nd the Rditive Trnfer, = I (, ( d nd the olution of thi eqution i: I d or I (, I (, ex = I I = o nd ince = co θ, we hve I (, I (, ex = co θ Beer Lw Bouguer Lw Lmbert Lw.85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge 4 of 5

5 If we hve I(θ o, lowly vrying function of frequency,, nd We define the trnmiion function, t = ex co θ I, d = I θ, ex d co θ ( ( Δ Δ = I θ o, tδ A lot of work h been done to develo method for comuting thi men trnmiion nd we will return to thi toic nd exmine it in detil little lter. Ce II: Let u conider cloudle tmohere nd the infrred ortion of the electromgnetic ectrum. Due to the roximte ertion of the olr emiion ectrum nd lnetry emiion ectrum ( dicued reviouly by Prof. Prinn, we now hve: ( ( J B, And the Rditive Trnfer Eqution reduce to: (, ( = d I, B, ( Thi i liner firt order eqution. If we ly e following eqution: n integrting fctor, we obtin the e B d = = Ie e I e d d (3 Let conider the uwrd intenity t level, z ( >. The origin of oticl deth i t the to of the tmohere nd we will need to integrte from the level, z, to the urfce. It i therefore convenient to chnge the vrible of integrtion to ' = z we mut integrte over oticl deth rnging from zero to the oticl deth t the urfce of the erth. Thu, we hve Eq. 4: ' ( z ( z B ' B ' Ie = e d ' = e d (4 z z z nd finlly the olution we re eeking: z z z d I( z = I( e B e (5 where the ubcrit,, in Eq. 4 nd 5 refer to the urfce of the erth..85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge 5 of 5

6 And, t the ground we tyiclly hve nerly blck body emiion in the infrred, o I( cn be ε B i the Plnck Blck Body relced by B where ε i n emiivity (ner unity nd Function. And the downwrd olution i imilrly given by: z ( = ( d z I z B e (6 In thee eqution we ve droed the frequency ecifiction for imlifiction. But we ll need to kee in mind tht oticl deth, Plnck function nd rdition intenity lwy deend on frequency (or wvelength. If the temerture i known throughout the tmohere, n exct olution i oible: i.e. B = B T ( Conider reure coordinte for which d=α dz =kρ dz =kχ ρdz kχ d d in ce g g dz = = ρ where k i the bortion cro-ection (in cm /gm, ρ i the borber m denity nd χ i m mixing rtio of borber,. Now, let go bck to the forml olution of the R.T.Eq. (Eq. 5 nd exmine the detil: ( ( B B T ( z k χ e ex dp Trnmiion g z e ( z k χ ex z dp g.85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge 6 of 5

7 nd if the ground cn be conidered blck ( reonble umtion in much of the therml infrred, we hve εb (,T B (,Tg So finlly, we hve: χ ( = k I,Pz B Tg ex d g z ground contribution kχ χ k B( T ex d' d g g z z tmoheric contribution (7 For downwrd rdition I (,, we imilrly hve: z z kχ χ k I (,z = B( T ex d' d (8 g g Let u recll tht we defined the trnmiion function follow: k χ = g z t ex d From bove, we hve ( z ( I, = B T dt t where T = T(t (9 with imilr exreion for the uwrd intenity: t ( = I, B T dt B T t ( z where t = trnmiion In rctice moleculr bortion by tmoheric ge (H O, CO, O 3, N O, CO, CH 4, O, etc. B,T. fluctute ridly with reect to frequency comred with the Plnck function Therefore, when we conider ectrl intervl rorite for meurement, we cn tke the frequency vrition into ccount nd we hve: Δ Δ t ( z I = I, d = B,T dt where = Δ z k χ d' t ex d g (.85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge 7 of 5

8 nd thi frequency verged trnmiion rereent the object of mny yer of work in tmoheric rditive trnfer by mny eole. We will dicu thi more comletely when we dicu the HITRAN & MODTRAN trnmiion nd rdition model lter. Ce III: Let u go bck to Eq. 9 nd conider the ource function under condition when the olr nd diffue rdition field i much greter thn the Plnck emiion. The roximte ertion of olr rdition nd lnetry emiion will gin be invoked to exmine the rditive trnfer F θ >> B, roblem in the viible ortion of the ectrum where nd the cttered rdition, I ( θ', φ, = >> B (, A before, the forml olution i: ( ( ( ' I,,, φφ, = J ',,, φφ, e d ' > ( ( ( ' I,,, φ, φ = J ',,, φ, φ e d ' < ω where J ( ',,, φφ, = P ( φ,, ', φ' I ( ', φ,, ', φ' d ' d φ' 4π F ω ex P (, φ,, φ 4π ( The olution to thi roblem require knowledge of the ditribution of ctterer, the oticl roertie of the ctterer nd their Phe Function (the robbility tht hoton incident from rticulr direction will be cttered into nother ecific direction. In generl, we mut lo del with the comlex roblem of multile cttering. We ll invetigte the roce of Mie Scttering nd Abortion by hericl rticle, hving ecified ize nd oticl roertie. We ll ue comuter rogrm tht rovide exct olution for Mie Scttering nd Abortion. Then, we ll be deling with comuter model cble of comuting the rdition field for multile cttering, uing rocedure known Dicrete Ordinte which divide the rdition field into Fourier comonent nd integrte the et of indeendent eqution uing Guin qudrture.. Rdition Intenity nd Rdition Flux. Rdition Intenity mount of energy er unit time contined in n element of olid ngle which flow through cro ection of unit re erendiculr to the direction of the bem. Let u conider tht we hve iotroic rdition of intenity I o flling on one fce of horizontl lb:.85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge 8 of 5

9 Solid Angle (in hericl coordinte b. Rdition Flux in Verticl Direction mount of energy er unit time croing unit urfce erendiculr to the z direction. π π π π F = I in θ dθ dφ = I in θ co θ dθ dφ o π = πi coθ inθdθ = πi.85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge 9 of 5

10 For Iotroic Rdition, the flux i π time the intenity of tright bem. Aliction uing Rdition Intenity. Remote ounding. Stellite meurement 3. Trget detection over horizontl/verticl th Aliction uing Flux:. Heting/Cooling of tmohere. Rdition effect on climte. 3. Aroximte Solution for Plnetry Rdition From Eq., we hve: = I d B (3 Thi cn be trnformed into n integrl eqution by integrting both ide over ll ngle: π π d ddφ = I ddφ 4πB d π I d = π I d 4πB d divergence of net totl flux totl flux in uwrd flux ( 4π I n LTE encloure d d ( πf If I = I = con tn t Id = I leding to the rditive trnfer eqution in net-flux form: df 4d = I B (4 Now multily both ide by nd integrte overe ll ngle: π π π d d I d d B d v φ = φ ddφ d π I d = d π K π F net uwrd flux.85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge of 5

11 dk d = F (5 Now, I look omething like thi In order to olve thee eqution, Eddington rooed two-trem roximtion: ( I I Thu I > π 4π I = I ddφ π I I π ( I π F = I ddφ π I (6 (6b π K I d d ( I I 3 π = φ π (6c df = ( I I B 4d (6d where we hve ued Eq. 4. d ( I I = F 3d (7 where we hve ued Eq. 6c..85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge of 5

12 Or, differentite Eq. 6d nd ue Eq. 7: df d db ( I I = 4d d d df 3 db = F 4d 4 d df d db 3F = 4 (8 d Thi i known Eddington Eqution. The two required boundry condition re uully given in the form of I - or I. We hve from Eq. 6d. df I = B I 4 d df B I F ( = 4d Uing 6d where we hve mde ue of Eq. 6b. df Thu: I = B 4d F = E B (t bottom eg cloud-to or urfce ( = = F t to ince I = t nd I = I F df = B 4d F (From I eqution bove = (t to ince no downwrd diffue rdition To rovide ome imle nlyticl olution it i ueful to conider the grey roximtion wherein α i relced by α (= grey bortion coefficient..85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge of 5

13 Thu, eqution of rditive trnfer cn be integrted over frequency ince d = α dz d = αdz which i now indeendent of. Uing the nottion = d = I J d df 3F = d db 4 d ( df I = B F = E B t bottom or F t to 4d df I = B F ( = t 4d 4 ST to. Alo note tht B = from erlier lecture. π where =Stefn contnt. Exmle: Suoe we hve n tmohere t ret (i.e. no dynmicl or ltent het fluxe. Suoe lo tht net rditive heting i zero everywhere tht i the net uwrd flux π F = contnt (i.e. non-divergent. Thi tte i clled rditive equilibrium. Eddington eqution i now: F = 4dB 3d I = B F = F t to I = B F = t to or B = F t to.85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge 3 of 5

14 = i. e. B F β( 3 db = F d 4 β 3 B( B( = F 4 B( F = 3 4 F B( = F 3 4 At the to of the tmohere we need for the lnetry verge: net incoming olr flux = net outgoing lnetry flux ( A Sπ 4π =πf F = ( A S 4π A S 3 B ( = 4π 4 4 (T A S 3 A S 4 3 = π 4π T = Thi i the imlet exreion of the greenhoue effect..85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge 4 of 5

15 ( Note:. Sometime A S i written T 4 e where T e i clled the effective temerture 4 of the lnet. (T e = 54.K for Erth.. In rditive equilibrium, the temerture of the urfce i not equl to the temerture of the ir immeditely bove the urfce. In rticulr t z= (or = : I = B( F = EB EB from reviou ge where B 3 EB = B( F= F F 4 = F = F 4 ( ( A S 4 T 3 = 4E 4 urfce tem. in rditive equilibrium for the urfce temerture in rditive equilibrium. For the erth, let u tke: ex ( z E 4 A =.3 S =.35 x 6 erg cm - ec - h = cle ht. of rincil = 5.67 x -5 erg cm - deg -4 ec - tmoheric borber (H O km. h And we obtin: T = 359.3K Wht i tem. of tmohere ner urfce? Wht i temerture grdient of tmohere t urfce? Wht i temerture t to of tmohere?.85, Atmoheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Ronld Prinn Pge 5 of 5