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.85, Atmospheric Rdition Dr. Robert A. McCltchey nd Prof. Sr Seger. Eqution of Rditive Trnsfer Specific Intensity of Rdition I,, E Energy da Are d Solid Angle d frequency int ervl dt time In n intervl ds, we lose intensity by extinction (scttering nd bsorption) nd gin it by emission nd scttering. Lmbert s Lw: The extinction process is liner, independently in the intensity of rdition nd in the mount of mtter, provided tht the physicl stte (i.e. temperture, pressure, composition) is held constnt. From Lmbert s Lw, the chnge of intensity long pth ds is proportionl to the mount of mtter in the pth nd to the intensity of rdition: di extinction losses I ds where = volume extinction coefficient () The rgument tht the extinction process is liner in the mount of mtter pplies with equl force to the emission process. Therefore, we write: di gins J ds () where we hve defined the source function, J. The extinction coefficient cn be expressed s the sum of n bsorption coefficient k nd scttering coefficient. k (3).85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge of 5
The most generl problem in tmospheric rdition, therefore, hs source function consisting of two prts, J k J therml J scttering (4) where k J therml 4 for isotropic emission. J sct, on the other hnd, is given by two terms, describing the diffusely scttered rdition nd the singly scttered incident bem of rdition (the sun). di ds,, I,,,, ' ' d P,,,' ' I,, (5) 4 F exp cos 0 P,, 0, 0 4 where P,,,' ' is the scttering phse function (or scttering digrm) nd is normlied d such tht P where d is n element of solid ngle. The shpe of the phse function cn 4 4 d be usefully chrcteried by single number, cos (cos )p where is the scttering 4 4 ngle nd <cos > is clled the symmetry prmeter (which vries between nd - nd is 0 for isotopic scttering. Dividing by, we hve: di,, I,, ds ' ' P,,,' ' I,, sin 'd 'd ' 4 0 0 0 F exp cos P,,, 4 (6) Let us now introduce the following definitions: d d d ds cos (7) where = verticl opticl depth mesured from the top of the tmosphere 0t. Note tht this coordinte differs for ech..85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge of 5
We then obtin the following Rditive Trnsfer Eqution in differentil form in plne prllel tmosphere: di,, ' ' ' ' ' ' I,, P,,, I,, sin d d d 4 F exp cos P,,, 0 0 0 4 or di,, I,, J,, d where we hve defined the source function, J,, s: (8) ' ' ' ' ' ' J,, P,,, I,,,, dd 4 F exp 0 P,,, 0 0 4 (9) nd = = Single Scttering Albedo. Let us lso be reminded tht the tmosphere in generl contins both gses nd prticultes (erosols). Ech hs scttering nd bsorption properties tht we need to consider. Thus we hve: k gses k erosols gses erosols There re mny cses where the physicl sitution enbles the neglect of one or more of these terms nd relted simplifiction of the Rditive Trnsfer Eqution. We will now exmine few such cses. Let us first pply Kirchoff s Lw to our Rditive Trnsfer Eqution. The rtio of emission nd frctionl bsorption in ny direction of slb of ny thickness in thermodynmic equilibrium equls the blck body intensity. So in non-scttering tmosphere, we hve,, 4 B Blck Body Function B T 3 h c e h kt B T hc e 5 hc kt where B is isotropic.85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge 3 of 5
This gives us: ' ' ' ' 4 J,, B P,,, I,,,, dd 4 F exp P,, 0, 0 0 (9) Also if ll relevnt properties ( imuth ngle, i.e. I,, I, nd we hve:,,b ) re horiontlly invrint, then I is independent of di, I, J (0) d Cse I: Let us first consider the simplest cse chrcteried s follows: ) We observe sunlight of visible wvelengths through non-scttering tmosphere: F B 4 nd lso B I, or b) Bright, rtificil source of rdition shining through n tmospheric pth (e.g.- lser): We then hve tht J Eqution is simplified to: I, nd the Rditive Trnsfer di, I, () d nd the solution of this eqution is: I di d or I 0, I 0 0,0exp I I 0 o 0 0 nd since 0 cos 0, we hve I 0, I 0 0,0exp cos 0 Beer s Lw Bouguer s Lw Lmbert s Lw.85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge 4 of 5
If we hve I( o, 0) s slowly vrying function of frequency,, nd We define the trnsmission function, t exp cos 0 I, d I 0,0 exp d cos 0 I o,0 t A lot of work hs been done to develop methods for computing this men trnsmission nd we will return to this topic nd exmine it in detil little lter. Cse II: Let us consider cloudless tmosphere nd the infrred portion of the electromgnetic spectrum. Due to the pproximte seprtion of the solr emission spectrum nd plnetry emission spectrum (s discussed previously by Prof. Seger), we now hve: J B, And the Rditive Trnsfer Eqution reduces to: di, d I, B, () This is liner first order eqution. If we pply e s following eqution: n integrting fctor, we obtin the di e B d Ie e I e d d (3) Lets consider the upwrd intensity t level, ( > 0). The origin of opticl depth is t the top of the tmosphere nd we will need to integrte from the level,, to the surfce. It is therefore convenient to chnge the vrible of integrtion to ' s we must integrte over opticl depths rnging from ero to the opticl depth t the surfce of the erth. Thus, we hve Eq. 4: τs τs -τ ' B( ') -τ ' μ τ τ -B( ) s μ τ ( τ τ )/ μ τ e τ μ τ μ τ Ι e = e d τ'= d (4) nd finlly the solution we re seeking: s s d I Ise B e (5) where the subscript, s, in Eqs. 4 nd 5 refers to the surfce of the erth..85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge 5 of 5
And, t the ground we typiclly hve nerly blck body emission in the infrred, so I( s ) cn be B is the Plnck Blck Body replced by B where is n emissivity (ner unity) nd Function. s And the downwrd solution is similrly given by: 0 d I B e (6) In these equtions we ve dropped the frequency specifiction for simplifiction. But we ll need to keep in mind tht opticl depth, Plnck function nd rdition intensity lwys depend on frequency (or wvelength). If the temperture is known throughout the tmosphere, n exct solution is possible: i.e. B B T p s Consider pressure coordintes for which dd k d k d k dp dp sin ce g g d where k is the bsorption cross-section (in cm /gm), is the bsorber mss density nd is mss mixing rtio of bsorber,. Now, lets go bck to the forml solution of the R.T.Eq. (Eq. 5) nd exmine the detils: B BTp ps s k e exp dp Trnsmission g p e p k exp p dp g.85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge 6 of 5
nd if the ground cn be considered blck ( resonble ssumption in much of the therml infrred), we hve B,T B,Tg So finlly, we hve: s k I,P B Tg exp dp g p p ground contribution ps p k k BTp exp dp' dp g g p p tmospheric contribution (7) For downwrd rdition I,, we similrly hve: p p k k I,p BTpexp dp' dp (8) 0 g p g Let us recll tht we defined the trnsmission function s follows: k g p p t exp dp From bove, we hve p I,p B T dt t where T = T(t) (9) with similr expression for the upwrd intensity: t s I,p B T dt B T t (0) where t = trnsmission s s In prctice moleculr bsorption by tmospheric gses (H O, CO, O 3, N O, CO, CH 4, O, etc.) fluctute rpidly with respect to frequency compred with the Plnck function B,T. Therefore, when we consider spectrl intervl pproprite for mesurement, we cn tke the frequency vrition into ccount nd we hve: t I I,p d B,T dt where p k dp' t exp d p g ().85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge 7 of 5
nd this frequency verged trnsmission represents the object of mny yers of work in tmospheric rditive trnsfer by mny people. We will discuss this more completely when we discuss the HITRAN & MODTRAN trnsmission nd rdition models lter. Cse III: Let us go bck to Eq. 9 nd consider the source function under conditions when the solr nd diffuse rdition field is much greter thn the Plnck emission. The pproximte seprtion of solr rdition nd plnetry emission will gin be invoked to exmine the rditive trnsfer F B, problem in the visible portion of the spectrum where nd the scttered rdition, I ',, B, As before, the forml solution is: 0 0 0 0 0 ' I,,,, J ',,,, e d ' 0 0 0 0 0 0 ' I,,,, J ',,,, e d ' 0 where J ',, 0,, 0 P,, ', ' I ',,, ', ' d ' d ' 4 F exp P,, 0, 0 4 0 () The solution to this problem requires knowledge of the distribution of sctterers, the opticl properties of the sctterers nd their Phse Function (the probbility tht photon incident from prticulr direction will be scttered into nother specific direction). In generl, we must lso del with the complex problem of multiple scttering. We ll investigte the process of Mie Scttering nd Absorption by sphericl prticles, hving specified sies nd opticl properties. We ll use computer progrm tht provides exct solutions for Mie Scttering nd Absorption. Then, we ll be deling with computer model cpble of computing the rdition field for multiple scttering, using procedure known s Discrete Ordintes which divides the rdition field into Fourier components nd integrtes the set of independent equtions using Gussin qudrture.. Rdition Intensity nd Rdition Flux. Rdition Intensity mount of energy per unit time contined in n element of solid ngle which flows through cross section of unit re perpendiculr to the direction of the bem. Let us consider tht we hve isotropic rdition of intensity I o flling on one fce of horiontl slb:.85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge 8 of 5
Solid Angle (in sphericl coordintes) b. Rdition Flux in Verticl Direction mount of energy per unit time crossing unit surfce perpendiculr to the direction. Let us integrte over the upper hemisphere. F I sin d d I sin cos d d 0 0 0 0 o I cos sind I 0 0 0.85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge 9 of 5
For Isotropic Rdition, the flux is times the intensity of stright bem. Applictions using Rdition Intensity. Remote sounding. Stellite mesurements 3. Trget detection over horiontl/verticl pths Applictions using Flux:. Heting/Cooling of tmosphere. Rdition effects on climte. 3. Approximte Solution for Plnetry Rdition (See Goody & Yung, pp. 57-63). From Eq., we hve: di I d B (3) This cn be trnsformed into n integrl eqution by integrting both sides over ll ngles: di d dd I dd 4B 0 0 d I d I d 4B d divergence of net totl flux totl flux in upwrd flux 4 I n LTE enclosure d d F If I I0 cons tn t Id I leding to the rditive trnsfer eqution in net-flux form: df 4d I B (4) Now multiply both sides by nd integrte over ll ngles: di d d I d d B d 0 0 v 0 0 dd 0 d I d d F net upwrd flux K.85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge 0 of 5
dk d F (5) Now, I looks something like this In order to solve these equtions, Eddington proposed two-strem pproximtion: I I 0 Thus I 0 4 I I dd I I 0 0 F I dd I I (6) (6b) K I d d I 3 0 I (6c) df I I B 4d (6d) where we hve used Eq. 4. d I I F 3d (7) where we hve used Eq. 6c..85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge of 5
Or, differentite Eq. 6d nd use Eq. 7: df d db I I 4d d d df 3 db F 4d 4 d df d db 3F 4 (8) d This is known s Eddington s Eqution. The two required boundry conditions re usully given in the form of I - or I +. We hve from Eq. 6d. df I I 4 d B df B I F 4d Using 6d where we hve mde use of Eq. 6b. Thus: I df B F 4d E s s B (t bottom eg cloud-top or surfce) nd I I F F t top since I 0 t 0 df B 4d F (From I + eqution bove) = 0 (t top since no downwrd diffuse rdition) To provide some simple nlyticl solutions it is useful to consider the grey pproximtion wherein is replced by (= grey bsorption coefficient)..85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge of 5
Thus, eqution of rditive trnsfer cn be integrted over frequency since d d d d which is now independent of. Using the nottion d di I J d df 3F d db 4 d df I B F EsBs t bottom or F t top 4d df I B F 0 t 4d 4 ST top. Also note tht B from erlier lecture. where s=stefn s constnt. Exmple: Suppose we hve n tmosphere t rest (i.e. no dynmicl or ltent het fluxes). Suppose lso tht net rditive heting is ero everywhere tht is the net upwrd flux F = constnt (i.e. non-divergent). This stte is clled rditive equilibrium. Eddington s eqution is now: F 4dB 3d I B F F t top I B F 0 t top or B F t top.85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge 3 of 5
i.e. B 0 F 3 db F d 4 0 0 3 B B0 F 4 B F 3 4 F B F 3 4 At the top of the tmosphere we need for the plnetry verge: net incoming solr flux = net outgoing plnetry flux A S 4 F F A S 4 B A S 3 4 4 4 s T A S 3 4 4 A S 4 3 T 4s 4 Where s = Stetn-Boltmnn constnt. This is the simplest expression of the greenhouse effect..85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge 4 of 5
Notes:. Sometimes A S is written s T 4 e where T e is clled the effective temperture 4s of the plnet. (T e = 54.K for Erth).. In rditive equilibrium, the temperture of the surfce is not equl to the temperture of the ir immeditely bove the surfce. In prticulr t =0 (or = s ): I B s F s s s s EB EB from previous pge where B F 3 4 3 EB s s Bs F F s F 4 3 F s 4 A S 4 T 3 s 4sE 4 s s surfce temp. in rditive equilibrium for the surfce temperture in rditive equilibrium. For the erth, let us tke: s exp Es s 4 A = 0.3 S =.35 x 0 6 erg cm - sec - h = scle ht. of principl s = 5.67 x 0-5 erg cm - deg -4 sec - tmospheric bsorber (H O) km. h And we obtin: T s = 359.3K Wht is temp. of tmosphere ner surfce? Wht is temperture grdient of tmosphere t surfce? Wht is temperture t top of tmosphere?.85, Atmospheric Rdition Lecture Dr. Robert A. McCltchey nd Prof. Sr Seger Pge 5 of 5