Atmospheric Radiation Fall 2008

Transcription

1 MIT OpenCourseWare Atospherc Radaton Fall 8 For nforaton about ctng these aterals or our Ters of Use, vst:

2 .85, Atospherc Radaton Dr. Robert A. McClatchey and Prof. Sara Seager 3. Scatterng of Radaton by Molecules and Partcles a. Introducton Here, we ll deal wth wave aspects of lght rather than quantu aspects. Consder coponents of electrc feld n utually perpendcular drectons, parallel and perpendcular to the plane of propagaton and propagatng n the z drecton: ( tkz) Er are crcular frequency ( tkz) E ae k () The ntensty s gven by: I E E E E a a * * r r r () Let us frst consder Sngle Scatterng. We ay consder a sngle partcle or a sall volue of partcles such that tterng events wll all be sngle tterng events. Fg. p = phase atrx p = phase functon dv P I dv I k P I or I k 4R 4R k = tterng cross-secton per unt volue k, [k ] N length dv dv, [ ] length N = total no. of partcles Q, where G geoetrcal cross sec ton G.85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 6

3 , Q Q Scatterng effcency G G P s the phase atrx and provdes the angular dstrbuton and polarzaton of the ttered lght. For our purpose here, lets consder the total ntensty of the radaton whether polarzed or not. Then the ter,, s the phase functon or tterng dagra whch defnes the probablty for tterng of unpolarzed ncdent lght n any drecton. p s noralzed such that: p p d where d eleent of sold angle 4 (3) Let us defne <cos > = ). cos p d 4 where s the tterng angle (see Fg. <cos > = ansotropy paraeter (or asyetrc paraeter) and can vary between + and -. <cos > = for sotropc tterng. We defne analogous ters for partcle absorpton and we have: k k k ext abs ext abs Q Q Q ext abs Then, we defne: rq (4) k Q sngle tterng albedo k Q ext ext ext For practcal applcatons, k and can be taken as constants and P s a functon only of the tterng angle,. Ths specal case s vald for: () randoly orented partcles, each of whch has a plane of syetry () randoly orented asyetrc partcles, f half the partcles are rror ages of the others. (3) Raylegh tterng and Me tterng. Another portant defnton: a where a = partcle radus. nr n where nr = real part of the ndex of refracton and n = agnary part. n s responsble for absorpton and s responsble for tterng. n r.85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 6

4 For water, n r =.33 across the vsble and near nfrared. And n wll depend on the knds of aterals dssolved n the water drops. It wll therefore be uch ore of a functon of wavelength. Returnng to the analytcal expresson for the electrc feld as a bea of radaton passng through a sngle partcle: E(z, t) E e ( t kz) The ntensty of radaton vares as the square of the E-feld. So, we have: ( tkz) I E e k and So, we have : E(z, t) E e z (nrn ) t E e zn znr t e and I E e 4zn 4znr t e and usng the defnton of sze paraeter daeter of drop, we have: a where we wll take z = a = I E e e 4n ( 4n r zt) absorpton (5) b. Me tterng stll sngle tterng. E s Er exp( kr kz) S (, )S 4(, ) r s kr S E 3(, )S (, ) E (6) at dstance R n the far feld If we consder sotropc, hoogenous, spheres, we have: S( ) S I S ( ) k R And we have the transforaton atrx: F I The S values are n general coplex nubers and functons of tterng angle..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 3 of 6

5 F * * * * SS SS SS SS * * * * SS SS SS SS SS SS SS SS SS SS SS SS * * * * * * * * whch s proportonal to the phase atrx: The noralzaton condton on P leads to: F CP (8) C 4 F d 4 and snce = effectve cross secton, we have: IR d and I 4 C 4 F Iod k 4 4 where we ve used I= F k R I o So we have: k p * * F (SS SS) 4 defnes phase functon F k p 4 * * (SS SS) (9) k p * * 43 k p * * F (SS SS ) F (SS SS) 4 4 For a sngle sphere, we have: S n ann nn(n ) bn n S n bnn nn(n ) an n ().85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 4 of 6

6 n and n are functon only of and relate to Legendre Polynoals. a n and b n functons. are functons of a x and nr n and nvolve Sphercal Bessel We also have: Q (n )(a a b b) * * n n n n x n Q (n) R (a b) () ext e n n x n 4 n(n) * * n * cos R e(ana n bnb n) R e(anb n) n n(n) x Q n All above s for Sngle Scatterng fro a Sngle Sphere. In general, f optcal thckness s not too large, sngle tterng can be appled to a dstrbuton of partcles assued to be ndependent. We then have: r r r r k (r)n(r)dr r Q (r)n(r)dr r r ext ext ext r r () k (r)n (r) dr r Q (r)n (r) dr where n(r) = sze dstrbuton, descrbng the nuber of the partcles havng rad between r and r+dr over the range r to r..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 5 of 6

7 c. Geoetrc Optcs: When r >>, we can use ray theory of lght due to Fresnel (see Van de Hulst, Ch. 3, Lght Scatterng by Sall Partcles). Ternology s: dffracton external reflecton double refracton 3 frst ranbow 4 second ranbow and f P( ) P ( ), then P ( ) sn dd, s for non-absorbng spheres fro Fresnel theory. 4 real =.33 =..5.5 always true n geoetrc optcs often suffcent to consder just these For non-absorbng partcles, dffracton = \ of ttered lght. Thus, the geoetrc optcs ltng value of Q ext =...85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 6 of 6

8 d. Raylegh Scatterng: r and r where nr n Partcle can be consdered to be n hoogenous external electrc feld Radaton penetrated partcle quckly. partcle own feld s neglgble n the process. E k pencdent kn where angle between dpole sne R oent and drecton of tter (3) 3 ( cos ) 3 sn sn ( cos ) 4 p( ) 3 cos 3 cos (4) p 3 ( cos ) (5) 4 whch gves angular dstrbuton of ntensty ttered by sall partcles the Raylegh tterng. We also obtan: 8 4 Q x Qabs 4xI (6) 3 note that Q Q as x abs note 4 th power of x or -4 dependence Ths result s the sae as Me tterng as ltng case as x (r<<)..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 7 of 6

9 .85 Lecture Notes (Atospherc Radaton) Multple Scatterng Refer back to Eq. fro the frst set of Atospherc Radaton lecture notes where we dscussed Case III whch arses due to the followng two condtons: F B (T) () I(,, )>>B ( ) () The resultng equaton of transfer s: di (,, ) I(,, ) J(,, ) d (3) where / o J (,, ) P(,, ', ') I ( ', ', )sn 'd 'd ' e P(, )F 4 4 and the foral soluton s: d ' ( ')/ I(,,,, ) J( ',,,, )e s ( ' )/ d ' I(,,,, ) J( ',,,, )e (4) Due to the coplextes of evaluatng the ntegrals n Eq. 4, a nuber of technques have been used to generate nuercal results:. Dscrete Ordnates. Doublng or Addng Method 3. Successve Orders of Scatterng 4. Iteraton of Foral Soluton 5. Invarant Ebeddng 6. Method of X and Y Functons 7. Sphercal Haroncs Method 8. Expanson n Egenfunctons 9. Monte Carlo Method We wll focus soe attenton on the Dscrete Ordnates Method and apply an avalable coputer progra to soe exercses..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 8 of 6

10 Radatve Transfer n a Scatterng Atosphere. Coordnate syste n a plane parallel atosphere Here poston defned by z (or ) only. Recall that optcal depth related to alttude z by d = -dz where s the extncton coeffcent. cos = ; = nclnaton to upward outward noral Notaton cos o = o ; o = nclnaton to upward outward noral.85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 9 of 6

11 cos = ; = nclnaton to upward outward noral Fro sphercal geoetry, the cosne of the tterng angle, can be expressed n ters of the ncong and outgong drectons n the for: cos ' ' cos ' (5) Let us now dgress for a oent and exane the propertes of Legendre polynoals (whch coe to play n a varety of ways n radatve transfer probles). We ay consder wrtng the phase functon n ters of Legendre polynoals n the for: P(cos ) = N C P (cos ) (6) Legendre polynoals have the followng for, and orthogonal and recurrence propertes: d n!d n n P ( ) P n( ) ( ) (n,...) n n So, 3 P ( ) P ( )... P( )P( k )d = k k (7) P(h) P( ) P( ) Usng Eq. 5, the Phase Functon defned above ay be wrtten as follows: N P(,,, ) C P ( ) ( ) cos ( ) (8) Fro the orthogonalty condton, the expanson coeffcents are gven by: C P ( ) P ( )d,...n where we note that the phase functon s noralzed to unty:.85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 6

12 4 P( )dd There s an addton theore for Legendre polynoals whch allows us to wrte the Phase Functon as follows: N N P(,,, ) C P ( )P ( ) cos ( ) (9) where P denotes the Assocated Legendre polynoals and:! C (,)C l,,...n n! (),,, otherwse In vew of the expanson of the phase functon, the dffuse ntensty ay also be expanded n a cosne seres n the for: N I(,, ) I (, )cos( ) () Substtutng Eqs. 9 and nto Eq. 3, and usng the orthogonalty of the assocated Legendre polynoals, the equaton of transfer splts nto (N+) ndependent equatons, and ay be wrtten as: N di (, ) I (, ) (,) C P ( ) () d 4 x P ( )I (, )d C P ( )P ( )F e 4 N =,, N Let us rewrte these equatons as follows: di (, ) d I (, ) J (, ) (3).85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 6

13 wth the source functon gven by: J (, ) ( ) C P ( ) P ( ) I (, )d N, 4 4 N C P ( )P ( )Fe (4) To proceed wth the soluton of Eq. 3, we frst dscretze the equaton by replacng wth (= -n,., n, wth n=,,.) and the ntegral wth a su wth the weghts, a j n jn f( )d f( )a (5) j j The hoogeneous soluton for the set of frst-order dfferental equatons ay be wrtten: n k j j j (6) jn I (, ) L ( )e where and L j j ( ) and kj denote the egenvectors and egenvalues, respectvely, are coeffcents to be deterned fro approprate boundary condtons. On substtutng Eq. 6 nto the hoogeneous part of Eq. 3, the egenvectors ay be expressed by ( N n,) j q q j q 4( jk j ) qn ( ) C P ( ) a P ( ) ( ) The partcular soluton ay be wrtten n the for (7) I (, ) Z ( )e p (8) Fro Eq. 3, we have N Z ( ) C P ( ) j 4 (9) n F x aqp ( q)z ( q) P ( ) qn Equatons 7 and 9 are lnear equatons n j and z and ay be solved nuercally. The coplete soluton for Eq. 3 s the su of the general.85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 6

14 soluton for the assocated hoogeneous syste of the dfferental equatons and the partcular soluton. Thus, n kj j j jn I (, ) L ( )e Z ( )e () = -n,. +n In order to deterne the unknown coeffcents, L j, a q, boundary condtons ust be posed. In the dscrete-ordnates ethod for radatve transfer, analytcal solutons for the dffuse ntensty are explctly gven for any optcal depth. Thus the nternal radaton feld can be evaluated wthout addtonal coputatonal effort. And furtherore, useful approxatons can be developed fro ths ethod for flux calculatons. Advantages of Dscrete Ordnate Method a) In prncple - nuercal coputatons can be done for any order of approxaton. b) The nternal radaton feld s deterned - not just the Reflecton & Transsson. c) Accurate results (to about %) are achevable wth only a few streas (3-4) for ost cases. We wll utlze the Dscrete Ordnate coputer progra to do a few excercses..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 3 of 6

15 Multple Scatterng Coputatonal Technques. Dscrete Ordnates (We ll dscuss n detal n a few nutes.). Doublng or Addng Prncple: If reflecton and transsson s known for each of two layers, the reflecton and transsson fro the cobned layer can be obtaned by coputng the successve reflectons back and forth between the two layers. If the two layers are chosen to be dentcal, the results for a thck hoogenous layer can be bult up rapdly n a geoetrc (doublng) anner. 3. Successve Orders of Scatterng Prncple: Intensty s coputed ndvdually for photons ttered once, twce, three tes, etc. wth the total ntensty obtaned as the su over all orders. If the ntensty s expanded n a Fourer seres, the hgh frequency ters arse fro photons ttered a sall nuber of tes. Therefore, ost Fourer ters can be obtaned wth soe accuracy by coputng a few orders of tterng. 4. Iteraton of Foral Soluton Drect soluton of ntegral over source functon by dvdng atosphere nto layers wth sall optcal thckness. 5. Invarant Ibeddng Dfferental Equatons are developed whch gve the change of reflecton and transsson atrces when an optcally thn layer s added to the atosphere. It s a specal case of the Doublng or Addng technque. 6. Method of X and Y Functons Involves the deternaton of ntegral equatons for functons whch depend upon only one angle and are drectly related to Reflecton and Transsson atrces. The ntegral equatons need to be solved nuercally. The ntegral equatons are copletely specfed by a character functon dependng on the partcular phase functon. Ths ethod s due to Chandrasekhar. 7. Sphercal Haronc Method Intensty s edately expanded nto a fnte nuber of sphercal haroncs and then the Phase Functon s expanded n Legendre polynoals slar to the Dscrete Ordnate ethod. 8. Expanson n Egenfunctons Standard technque for solvng dfferental equatons. Fnd hoogenous soluton and partcular soluton. Apply boundary condton. Drect applcaton to coplete RTE s ponderous. Dscrete Ordnates technque depends on ths approach for solvng dscretzed set of equatons. 9. Monte Carlo Method Scatterng of an ndvdual photon can be consdered to be a stochastc process, wth the Phase Functon beng the probablty densty functon for tterng at a gven angle. Photons are allowed to play a gae of chance n a coputer and by recordng the hstory of a suffcent nuber of photons, the radaton feld can n prncple be deterned to an arbtrary accuracy. The basc splcty of ths ethod allows great flexblty, and hence t can be appled to coplcated probles whch would be vrtually nsoluble by other ethods..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 4 of 6

16 Isotropc Scatterng and Dscrete Ordnates Pertnent RTE: di(,, )/di(,, ) P(,,, ) I(,, )d d 4 e P(, )F 4 () For sotropc tterng, we have: P(,,, ) and I (, ) I(,, )d ().e. Intensty s azuthally ndependent. di(, ) I(, ) I(, )d F e d 4 (3) Applyng Gaussan Quadrature, and settng I = I (, ), we have: di I Ia Fe d 4 n j j jn n,..., n (4) Snce ths s lnear dfferental equaton, we need to seek the general soluton (soetes called the hoogenous soluton) and then the partcular soluton. Hoogenous soluton: k Try (guess) I g e where g and k are constants. di I I a (5) j j d j g( k) a g j j So, g ust be of the for L k where L s a constant. Substtutng ths back nto Eq. 5, we get the characterstc equaton for egenvalue k.85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 5 of 6

17 n a n j aj ( k) ( k ) jn j j j Note dfference n suaton (6) Ths Eq. has n roots, zero. General Soluton s: k =..n whch when = ncludes K values of n k L e I n,..., n (7) k Partcular Soluton: Try: I F he n,..., n 4 We have: n h h ah j j jn or h ah (8) n j j jn h ust be of the for = (9) wth n aj [ 4 j j () Addng the hoogenous and partcular solutons, we obtan: I n j k j Lj e F e kj 4 = -n,.. +n () The L j are deterned fro boundary condtons..85, Atospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 6 of 6