Atmospheric Radiation Fall 2008

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1 MIT OpenCourseWare Atmospherc Radaton Fall 008 For nformaton about ctng these materals or our Terms of Use, vst:

2 .85, Atmospherc Radaton Dr. Robert A. McClatchey and Prof. Sara Seager. Modelng Atmospherc Transmsson and Emsson Let us examne the low spectral resoluton structure of atmospherc absorpton (- transmsson) as ndcated n Fg.. The absorpton path here s the entre atmosphere lookng vertcally from the ground and from the km alttude level respectvely. otce how the absorpton propertes change wth alttude. Durng our dscusson of the Radatve Transfer Equaton, we found t necessary to defne a volume extncton coeffcent that s the sum of four terms: k gases k aerosols gases aerosols () These 4 terms nclude an absorpton coeffcent and a scatterng coeffcent for the gas molecules n the path and separately for the aerosols (partcles) n the path. In general, the coeffcents assocated wth aerosols and the scatterng coeffcent assocated wth molecules are relatvely slowng varyng functons of frequency (or wavelength) when compared wth molecular absorpton coeffcents. In ths segment of our course, we ll focus our attenton on the molecular absorpton coeffcents and examne methods that have been used to determne the related transmsson functons along atmospherc paths over whch pressure, temperature and molecular mxng ratos are changng. Fg. compares measurements and models of transmsson as a functon of frequency (wavenumber) over a horzontal sea level path 5. km long. Most of the actual molecular absorpton structure can be seen n these plots. (Compare Fg. wth Fg. ). a. HITRA (HIgh resoluton atmospherc TRAsmsson) Vbraton-rotaton transtons (and pure rotatonal transtons) of molecules wll lead to absorpton lnes assocated wth transtons between specfc energy levels of the molecule (See Fg. 3). Collsons modfy the energy levels of ndvdual molecules so that there s a dstrbuton of energy levels, effectvely smearng out the lne structure of absorpton as shown n Fg. 3. We wll refer to these features as lnes, however. Let us consder an ndvdual spectral lne assocated wth molecules located n the lower porton of the atmosphere (pressures hgher than about 0 mb). The Lorentz, pressure-broadened lne shape s gven by: k S T L P,T 0 P, T () where S T =Lne Strength, =Lne half-wdth =Lne center frequency L.85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 9

3 and S 0 k d We also have that the lne strength s gven by: n T 0 hce" T0 T ST ST0 exp T k T0 T (3) Rotatonal partton functon: n= for lnear molecules n= 3/ for tr-atomc non-lnear molecules where E = Energy of the lower state of the transton and T 0 s a standard temperature at whch S(T 0 ) s measured and P T 0 L L0 P0 T We have used the Boltzmann relaton to descrbe the number of molecules n the lower state of the transton. e QQ hce" kt v r And the lne strength s proportonal to ths number densty. S Pror to about 970, t was recognzed that the mportant atmospherc gases (e.g., H O, CO, O 3 ) had several seres of spectral lnes that could be modeled as beng ether regular or random over approprate spectral ntervals. Furthermore, most feld nstruments for radaton measurements were nadequate to measure the actual spectral structure anyway..85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 9

4 Therefore, band models were developed to represent the average transmsson (and absorpton) over lmted spectral regons. Ths led to the development of LOWTRA and MODTRA whch we ll dscuss later. By 970, expermental spectroscopy was mprovng and t was recognzed that a combnaton of theory and experment provded the opportunty to dentfy the spectral parameters of atmospherc molecules and thereby smplfy the computaton of atmospherc transmsson. Improvement n the speed and memory of computers also encouraged such a concept. From Eqs. & 3, we can see that f we defne the,s,,e" for each of several spectral lnes n the nterval,, we can compute the average transmsson (or absorpton) over the nternal wthout the need for a spectral model. (However, we do also requre a knowledge of the atmospherc temperature, pressure and molecular abundance.) We then have: exp k sngle lne of molecule (j) at frequency. m j for a exp k j mj j j exp k dmj j j and, fnally: ta t g 0 0 d, where g s an nstrument shape functon. g0d Summary of HITRA. HITRA 000 over 705,000 lnes of 38 gases coverng the spectral range: 0. m 0 50,000 cm - H O, CO, O 3, O, CO, CH 4 O, O, SO, O, H 3, HO 3, OH, HF, HCl, HBr, HI, ClO, OCS, H CO, HOCl,, HC, CH 3 Cl, H O, C H, C H 6, PH 3, COF, SF 6, H S, HCOOH, HO, O, ClOO, O +, HOBr.. HITRA 004 contans an even larger number of speces and lne parameters. 3. Lne strengths are proportonal to sotopc abundances. 4. Lne overlap from ndvdual molecular speces. 5. Lne overlap from dfferent molecular speces..85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 3 of 9

5 HITRA Complcatons. Lne wngs from outsde.. Lne shape. 3. Contnuum absorpton. 4. Pollutant lnes. We re now ready to examne the HITRA exercse labeled Exercse : HITRA-PC. Reference: R.M. Goody, Atmospherc Radaton, Oxford Clarendon Press, 964, Fg.. Fg..85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 4 of 9

6 Fg..85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 5 of 9

7 Fg. 3.85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 6 of 9

8 Atmospherc Radaton (.85) Band Models Scales of frequency. Planck functon slow varaton wth frequency for both Planck functon and ts dervatve.. Unresolved band contour Planck functon can be consdered constant over most ndvdual absorpton/emsson bands. 3. Spacng between rotatonal lnes ( 5 cm - ). 4. Monochromatc scale where absorpton coeffcent can be consdered constant (x0 - cm - for atmosphere pressure decreasng x0-4 cm - for Doppler lnes above about 30 km). Lne-by-lne (or convolved monochromatc) calculatons can be done, but remans formdable for practcal applcatons except for a few reference cases. Generally we stll need to make averages over many spectral lnes. Thus, we develop a varety of band models. So we need to deal wth average transmsson (and average absorpton). T T d () where s the wdth of the th frequency nterval and d T exp s the monochromatc transmsson. Because of the slow varaton of the Planck functon wth frequency, we can wrte the thermal emsson n terms of mean transmsson as follows: _ I I d d B e d d B dt () _ BdT 0 where B s the (almost constant) value of the source functon n the th nterval. Multplcaton Property of Transmsson:.85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 7 of 9

9 If we have sets of un-correlated spectra n a gven spectral nterval and f the lnes of ether array are randomly arranged n the nterval, the net transmsson n the nterval can be wrtten as: T, T T (3) Ths same property arses for regularly spaced sets of lnes n the spectral nterval,, where the lne spacng of the sets of lnes are dfferent. Ths condton cannot be exact as the spectral nterval gets suffcently small so that t contans too few lnes. But, for reasonable-sze ntervals contanng several lnes, Eq. 3 s a very good approxmaton. The physcs of molecular spectroscopy suggests that ths wll be so for any concevable set of condtons that may occur for atmospherc molecules. Sngle (Isolated) Lne of Lorentz Shape Here we wll lmt the dscusson to homogeneous paths (where T, p, and gas mxng ratos are constant along the path). Monochromatc Abso rpton: A T expk m (4) Integratng over all frequency space, we have: A exp k m d (5) where we also defne the Equvalent Wdth as +Δν / W A exp k m d (6) Δν / Ths term refers to the wdth of a rectangular lne whose center s completely absorbed, havng the same absorpton as the sngle lne under consderaton..85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 8 of 9

10 Usng the Lorentz lne profle of Eq. 7 and defnng varables, x & y: x Sm L tan y 0 L S L k (7) L y (8) A L exp x cosy d tan Integratng by parts yelds: A L L x (9) w th Lx xe x I x I x 0 (0) Ths functon s known as the Ladenburg and Reche functon where I x and I x are modfed Bessel functons defned as follows: n n xcos n n n cosnd 0 I x J x and J x e () For small x, the followng seres expanson s vald: 0 n n x L x x n n n! n n! () For large x, there s an asymptotc expanson: x 3. L x x n 3 n x n (3) n n! So for small x we have x Sm L A (4).85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 9 of 9

11 We could have obtaned ths result drectly from Eq. 5 by expandng the exponental: A m Sm k md k d We can see the condtons that need to apply -- km<< at all frequences (n partcular, at the lne center). For large x, we have: L A x Sm L (5) Sgnfcance of the Strong Lne Lmt: The center of the lne cannot be further absorbed after 00%, so ts only the lne wngs that can ncrease the absorpton. These lmts (Eqs. 4 & 5) are useful n the development of approxmatons for nfrared radatve transfer calculatons. Regular Band Model (Elsasser Model): Several atmospherc molecules demonstrate approxmate regular spacng of vbraton-rotaton absorpton lnes: e.g. CO, O, CO. Let us consder that all lne ntenstes are the same over a lmted spectral range. Then, the approprate absorpton coeffcent wll be: k S v L L where s the lne spacng (6).85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 0 of 9

12 Applcaton of Eq. 5 to ths absorpton coeffcent leads to: s A erf where erf x e dx x m x (7) 0 x S m For small x, we have A (8) whch s smlar to the ndvdual, solated lne n the strong lne lmt. The weak-lne Sm lmt wll be A Random Band Model (wth constant lne ntensty): Consder an array of dentcal lnes of the same molecular speces dstrbuted randomly n frequency between - and + where = mean lne spacng., The absorpton coeffcent s k and the resultng transmsson s: T expm k expmk (9) The probablty that a lne les n d s d / and the jont probablty that there are d lnes between and + d, and + d, n and n + d n s Consderng all possble arrangements of lnes, allowng each lne to le anywhere n the range to, the approprate average s: T d d exp mk (0).85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 9

13 + δ/ dν [ exp( mkν) ] Ν δ δ/ Τ= = +Νδ/ dν [ ( Ν exp( mk ν) )] Ν δ Νδ / () as, we have W T exp exp mk d exp () o r A exp A solated lne (3) The General Random Model: A frequency range cont ans lnes, each a sngle member of a dfferent nfnte, random array wth equal lne ntensty and lne spacng. From Eq., the mean transmsson of one of the array s: T exp W (4) The condtons for the multplcaton property are met for random arrays so we have the average transmsson for all arrays as: T T exp W (5) W exp (6) where W s the average equvalent wdth for the lnes n the frequency nterval. If we have a lst of lne ntenstes, we can compute W for a gven spectral nterval. The Correlated K-Dstrbuton: For a spectral nterval contanng several spectral lnes, the average transmsson s ndependent of the spectral locaton of the lnes, but depends on the absorpton coeffcents (or lne ntenstes, half-wdths) of the lnes. The Correlated K- Dstrbuton method for computng radatve transfer takes advantage of ths stuaton by groupng lnes accordng to the absorpton coeffcent k. Ths enables us to replace the usual frequency (or wavenumber) ntegraton for the average transmsson by an ntegraton over k-space. If the normalzed probablty dstrbuton functon for k n the nterval s gven by f(k) and ts mnmum and maxmum values are k mn and k max, respectvely, then the spectral transmsson can be expressed by Eq. 7:.85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page of 9

14 kmd km T m e e f k dk ( 7) 0 where we have set k mn 0 and k max, for mathematcal convenence and where 0 f k dk From Eq. 7, the probablty dstrbuton functon s the nverse Laplace transform, L -, of the spectral transmsson such that f k L T m (8) If the spectral transmsson can be expressed n terms of an analytc exponental functon and f the nverse Laplace transform can be performed, then an analytc expresson can be derved for the probablty dstrbuton functon. We can also defne a cumulatve probablty functon as follows: k gk fk dk (9) 0 where g(0) = 0, g(k) = and dg(k) = f(k) dk. By defnton, g(k) s a monotoncally ncreasng and smooth functon n k space. The spectral transmsson can now be wrtten as: kg kgj m T m e dg e g (30) M 0 j j From Eq. 9, snce g(k) s a smooth functon n k space, the nverse wll also be true here; that s, k(g) s a smooth functon n g space. Consequently, the ntegraton n g space, whch replaces the tedous wavenumber ntegraton, can be evaluated by a fnte sum of exponental terms as shown n Eq. 30. on-homogeneous Atmospheres: So far, all our band model analyses have homogeneous paths over whch temperature, pressure and molecular mxng rato are constant. In the real world, the decrease of pressure wth heght, atmospherc temperature profles and profles of molecular mxng ratos (such as water vapor) lead to substantal non-unformty n the vertcal and to a lesser extent also n the horzontal. Scalng approxmatons We pose the queston: How well can the transmsson of a varable atmospherc path be represented by the transmsson of a path at constant temperature and pressure? Snce temperature s less mportant n determnng atmospherc transmsson than pressure and molecular concentraton, t s customary to fx T by settng t equal to.85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 3 of 9

15 the average value for the path of ntegraton. If frequency can be separated from pressure and temperature n the monochromatc absorpton coeffcent, we can defne an exact scalng approxmaton: k P,T P,T (3) The optcal path then s P, Tdm F P, T path path P, T m where m P,T P,T dm path P, T P, T dm (3) (33) Ths would enable us to scale the amount of absorbng gas, m, and then calculate the absorpton coeffcent for the non-homogeneous path as f t were a homogeneous path of absorber amount, m. Unfortunately, the Lorentz profle doesn t even approxmately ft the form of Eq. 3, except for strong lnes. However, we have shown that specfc expressons are obtaned from weak lne and strong lne absorpton (See Eqs. 4 & 5). These same equatons can be wrtten for an optcal path n a non-homogeneous atmosphere. For weak lnes, we obtan: S T m S T dm path (34) and f we omt the temperature dependence, we obtan m dm (35) path For strong lnes, we have: path L, 0 L, 0 S T T P P m S T T P P dm (36) If we omt the temperature dependence, we obtan: m P path We can wrte both Eqs. 35 & 36 n the form, m path P P dm _ dm (37) P where = 0 for weak lnes and = for strong lnes..85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 4 of 9

16 Ths s the procedure that was used n the LOWTRA model n whch the parameter,, was determned emprcally from laboratory measurements coverng a range of homogeneous condtons coverng all atmospherc condtons of nterest. Returnng to Eq. 33, we can take both pressure and temperature dstrbutons nto account and obtan: where m T dm (38) T T S T From Eq. 36 we have a -parameter scalng law: φ(t) ΣS(T) m = = Σ pm= dm dm φ(t) S(T) o Σ { S(T) α (T) P/P dm} o [ L ] Σ{S(T) α(t)} (39) Ths s the so-called Curts-Godson approxmaton. on-homogeneous paths and the Correlated-k Technque Our correlated-k dstrbuton method has been developed for homogeneous paths. Its applcaton to non-homogeneous paths requres that we evaluate ts applcablty under condtons where the scalng approxmatons are vald (and for any other condtons that mght be representatve of the real atmosphere). The crcumstances under whch the Correlated-k technque gves exact results are: ) Whenever the condton dentfed n Eq. 3 s vald (for Strong Lorentz lnes and Doppler lnes); ) For weak absorpton, regardless of the lne shape; 3) For an solated lne, for an Elasser band (regardless of lne shape) and certan other lne shapes..85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 5 of 9

17 .85 Lecture otes MODTRA - 3-parameter model. - Based on HITRA data Average transmsson down to cm (uses a cm bn). - Frequency step = cm -. arrow-band models a) Random Model (Statstcal Band Model) Infnte array of spectral lnes wth Unform Statstcal Propertes. Statstcally smlar ntervals flank nterval of nterest. b) Regular Model (Elsasser Band Model) An almost regular array of lnes overlapped by un-correlated regular arrays.. Equvalent Wdth a) Sngle lne: km A e d b) Array of lnes: km A e d where =mean lne spacng Lets look at Random model n some detal: Consder array of dentcal lnes wth shapes descrbed by absorpton coeffcent, Let lnes be dstrbuted randomly between and. The absorpton coeffcent at the center of array s: k.85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 6 of 9

18 The transmsson s: t e xp m k exp mk d The pro bablty that a lne les n the nterval d s and the jont probablty that there are lnes between d an d d, and d...and and d s Consderng all possble arrangements of lnes n the range to the average transmsson s: t d exp mk d As ntegratons are all dentcal, we have: t exp mk d d expmk as, we obtan: W t exp exp mk d exp or A exp A solated lne ow consder a frequency range contanng lnes, each a sngle member of a dfferent nfnte array of random lnes of equal ntensty and lne spacng,. The mean transmsson of one of the array s: t exp W where W s the equvalent wdth of one lne n the array under consderaton..85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 7 of 9

19 The condtons for multplcaton property are met for random arrays and the arrays can be combned by multplcaton: W t t exp W exp where W s the average equvalent wdth for the lnes n the selected frequency range. ote that ths expresson was derved wthout reference to the lne profle and s not even restrcted to a sngle profle for all of the lnes. The only lmtaton s that the frequency nterval should be large enough for the multplcaton property to be vald. Based on statstcal band model for a fnte number of lnes n a spectral nterval, n w t = = where W Δν Mean Equvalent Wdth W km e d (6) and where n s the effectve number of lnes n the bn,. n where =average lne spacng F or large n, the transmsson smplfes to: t exp W / (7) However, n ths cm - bn of MODTRA, we cannot generally assume that n s large, so we must stck wth Eq. 6. Our problem then s to compute W and n for all spectral bns across the spectral range of nterest (vsble to mcrowave regon). Lets consde r that lnes don t overlap and agan we re assumng a Lorentz lne shape. We then have: km W e d yl u L m where y and u ave. spacng ( 8) L.85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 8 of 9

20 and L e I0 I where I 0 ( ) and I ( ) are Bessel Functons of the frst knd. For small, we have that L( )= and for large lmtng cases lead to the followng:, we have that L. These for small : m W S for large : m W S L Thus, there are three parameters necessary to defne the absorpton: L, S and S In the real world, fnte spectral bns and overlappng lnes modfy these parameters somewhat. The resultng MODTRA parameters are:. absorpton coeffcent = S T at several temperatures. lne densty = S S at several temperatures. average lne wdth = 3 L S S at a standard temperature These quanttes are easly calculated from the HITRA data base. A Curts Godson-type of approxmaton s appled to develop path weghted parameters for the usual n-homogeneous atmospherc paths over whch we wsh to perform radatve transfer calculatons. Benefts of MODTRA - More accurate - Covers wder range of atmospherc condtons - Useful at hgh alttude. Lablty of MODTRA Complexty of calculaton requres computer. Lets now turn to the PCModWn Exercse..85, Atmospherc Radaton Lecture Dr. Robert A. McClatchey and Prof. Sara Seager Page 9 of 9