Lecture 4. Beer-Bouger- Lambert law. Molecular (Rayleigh) scattering. Scattering and absorption by aerosol and cloud particles: Mie theory.

Transcription

1 Lectue 4. Bee-Bouge- Lambet law. Molecula (Raylegh) catteng. Scatteng and abopton by aeool and cloud patcle: Me theoy.. Bee-Bougue-Lambet law (Extncton law).. Bac of catteng. 3. Molecula (Raylegh) catteng. 4. Popete of aeool and cloud. 5. Scatteng and abopton by aeool and cloud patcle. 6. Remote enng applcaton baed on meauement of dect ola adaton (ee alo Lab 3). Requed eadng: S:.6; 4., 4.3; , 5.6, 5.7, Appendx ; Petty: 4, 7, Addtonal eadng S: 4.; 5.5 Advanced eadng Bohen, C. F., and D. R. Huffman, Abopton and catteng of lght by mall patcle. John Wley&Son, New Yok, pp. 53, Bee-Bougue-Lambet law (Extncton law). The fundamental law of extncton the Bee-Bougue-Lambet law, whch tate that the extncton poce lnea n the ntenty of adaton and amount of matte, povded that the phycal tate (.e., T, P, compoton) held contant.

2 Conde a mall volume V of nfntemal length d and unt aea A contanng optcally actve matte (gae, aeool, and/o cloud dop). Thu, the change of ntenty along a path d popotonal to the amount of matte n the path: d Fo extncton di ke I d, [4.a] I, I +??I Fo emon: di ka, Bd [4.b] d I whee κ e, the volume extncton coeffcent (LENGTH - ), whee κ a, the volume abopton coeffcent (LENGTH - ), and B the Planck functon. Hee the amount the patcle numbe concentaton. Recall that extncton due to catteng and abopton: k + e, k, ka, [4.] Integatng Eq.[4.a], we have I I, exp( k, ( ) d) I, exp( ) e [4.3] whee I, and I ae the ncdent and tanmtted ntente, epectvely. NOTE: optcal depth untle. NOTE: ame name : optcal depth optcal thckne optcal path Tanmon functon defned a T I I exp( ) [4.4] /, UNITS: tanmon functon untle (between and )

3 . Bac of catteng. Fgue 4. Smplfed vualzaton of catteng of an ncdent EM wave by a patcle. Conde a ngle abtay patcle conted of many ndvdual dpole. The ncdent electomagnetc feld nduce dpole ocllaton. The dpole ocllate at the fequency of the ncdent feld and theefoe catte adaton n all decton. In a cetan decton of obevaton, the total catteed feld a upepoton of the catteed wavelet of thee dpole, accountng fo the phae dffeence: catteng by the dpole coheent (.e., thee a defnte elaton between phae). Scatteng phae functon P(coΘ) defned a a non-dmenonal paamete to decbe the angula dtbuton of the catteed adaton 4π Ω P(coΘ) dω [4.5] whee Θ called the catteng angle between the decton of ncdence and decton of obevaton. Anothe fom of Eq.[4.5] π π ϕ (coθ)n ΘdΘd 4π P [4.6] 3

4 NOTE: The phae functon often wtten wth the notaton (θ', ϕ', θ, ϕ) o (µ', ϕ', µ, ϕ) (coθ) P( θ, ϕ, θ, ϕ) P [4.7] P (coθ) P( µ ', ϕ, µ, ϕ) whee (θ', ϕ') and (θ, ϕ) ae the phecal coodnate of ncdent beam and decton of obevaton: θ' and θ ae the zenth angle, and ϕ' and ϕ ae the azmuth angle; and µ co (θ) and µ' co (θ'). The catteng angle Θ expeed n tem of (θ', ϕ', θ, ϕ) a (ee S: Appendx ) co(θ) co(θ')co(θ) + n(θ')n(θ) co(ϕ'-ϕ) o co(θ) µ'µ + ( µ' ) / ( µ ) / co(ϕ'-ϕ) Fgue A. fom Stephen, Appendx, p.468 The aymmety facto g defned a g fo otopc catteng g P(coΘ)coΘd(coΘ) [4.8] 4

5 Fowad catteng efe to the obevaton decton fo whch Θ < π/: g> catteng n the fowad decton Backwad catteng efe to the obevaton decton fo whch Θ > π/: g< catteng n the backwad decton Scatteng doman: Raylegh catteng: π/ <<, and the efactve ndex m abtay (apple to catteng by molecule and mall aeool patcle) Raylegh-Gan catteng: (m ) << (not ueful fo atmophec applcaton) Me-Debye catteng: π/ and m ae both abtay but fo phee only (apple to catteng by aeool and cloud patcle) Geometcal optc: π/ >> and m eal (apple to catteng by lage cloud doplet, an, dop and ce cytal). 3. Molecula (Raylegh) catteng. Becaue the ze of atmophec molecule ae much malle than the wavelength of ola and IR adaton, catteng by atmophec gae the Raylegh catteng doman. In the Raylegh catteng appoxmaton, a molecule (o a mall patcle) condeed a an ndvdual, pont dpole. Conde a mall homogeneou phecal patcle (e.g., a molecule) wth ze malle than the wavelength of ncdent adaton E. Let p be the nduced dpole moment, then fom the clacal electomagnetc theoy we have whee α the polazablty of the patcle. p αe [4.9] 5

6 NOTE: Do not confue the polazaton of the medum wth polazaton aocated wth the EM wave. The catteed electc feld at the lage dtance (called fa feld catteng) fom the dpole gven (n cg unt) by E c p n( t γ ) [4.] whee γ the angle between the catteed dpole moment p and the decton of obevaton. In ocllatng peodc feld, the dpole moment gven n tem of nduced dpole moment by p p exp( k( ct)) [4.] and thu the electcal feld exp( k( ct) E E k α n( γ ) [4.] (Note hee k denote π/). Decompong the electcal vecto on two othogonal component pependcula and paallel to the plane of catteng (a plane contanng the ncdent and catteng beam), We have E E l exp( k( ct) E k α n( γ ) [4.3] exp( k( ct) El k α n( γ ) [4.4] E p Dpole γ E l Decton of ncdent adaton γ π/; γ π/-θ p l γ Θ Decton of catteng (out of page) 6

7 Ung that I c E Ω 4π [4.5] the pependcula and paallel ntente (o lnea polazed ntente) ae I 4 I k α / [4.6] I l 4 I lk α co ( Θ) / [4.7] Ung that the natual lght (ncdent beam) n not polazed (I I l I /) and that kπ/, we have I 4 I co ( ) π + I + I α Θ [4.8] l Eq.[ 4.8] gve the ntenty catteed by molecule (Raylegh catteng) fo unpolazed ncdent lght. Raylegh catteng phae functon fo ncdent unpolazed adaton 3 P (co( Θ)) ( + co ( Θ)) [4.9] 4 Eq.[ 4.8] may be ewtten n the fom 5 I 8π P( Θ) I (co( Θ)) α [4.] 4 3 4π Eq. [4.] may be ewtten n the tem of the catteng co ecton I P( Θ) I(co( Θ)) σ [4.] 4π Hee the catteng co ecton (n unt of aea) by a ngle molecule 5 8π σ α 4 [4.] 3 7

8 The polazablty of a pont dpole gven by the Loentz-Loenz fomula 3 m α [4.3] 4πN m + whee N n the numbe of molecule pe unt volume and mn κ n the efactve ndex. Fo a molecule n ola pectum: n about but depend on, and κ. Thu the polazablty can be appoxmated a α ( n ) [4.4] 4πN Theefoe, the catteng co ecton of an a molecule become σ 3 8π ( n ) f ( δ ) 4 [4.5] 3 N whee f(δ) the coecton facto fo the anotopc popete of a molecule, defned a f(δ) (6+3δ)/(6-7δ) and δ.35 Ung th catteng co ecton, one can etmate the optcal depth of the ente atmophee due to molecula catteng a top ( ) σ ( ) N( z) dz [4.6] NOTE: The Raylegh catteng co ecton (Eq.[4.]) and hence optcal depth ae nveely popotonal to the fouth powe of the wavelength (> blue colo of the ky) Appoxmaton of molecula Raylegh optcal depth (.e., optcal depth due to molecula catteng) down to peue level p n the Eath atmophee: whee the wavelength n µm. ( ).88 3 p mb [4.7] 8

9 Raylegh catteng eult n the ky polazaton. The degee of lnea polazaton Q Il I co Θ n Θ LP ( Θ) [4.8] I I + I co Θ + co Θ + l 4. Popete of aeool and cloud. Inteacton of patculate (aeool, cloud dop, ce cytal, an dop, etc.) wth electomagnetc adaton contolled by the patcle amount, ze, compoton (efactve ndex), and hape. Atmophec aeool ae old and/o lqud (o mxed phae) patcle upended n the a wth damete between about. µm to about 5- µm. Chemcal compoton: Indvdual chemcal pece: ulfate (SO - 4 ), ntate (NO - 3 ), oot (elemental cabon o black cabon), ea-alt (NaCl); mneal (e.g., quatz, SO 4, clay, feldpa, etc.) Mult-component (MC) aeool: complex make-up of many chemcal pece (called ntenally mxed patcle) Shape: Sphee: all aqueou aeool patcle (e.g., ulfate, ntate, etc.) Complex hape: dut, oot (.e., old patcle) 9

10 Clacal epeentaton of patcle ze pectum: Fgue 4. Idealzed chematc of the dtbuton of patcle uface aea of atmophec aeool patcle (fom Whtby and Cantell, 976). NOTE: fne mode (d <.5 µm) and coae mode (d >.5 µm); fne mode dvded on the nucle mode (about.5 µm < d <. µm) and accumulaton mode (.µm < d <.5 µm). The patcle ze dtbuton of aeool ae often appoxmated by a um of thee lognomal functon a N ln( /, ) N ( ) exp( ) π ln( σ ) ln( σ ) whee N() the patcle numbe concentaton, N the total patcle numbe concentaton of -th ze mode wth t medan adu, and geometc tandad devaton σ. [4.9]

11 Popety of the log-nomal functon: the k-moment k k N( ) d N exp( k (lnσ ) / ) [4.3] NOTE: Eq [4.3] help to quckly calculate the ma, volume, and uface aea ze dtbuton fom the patcle numbe ze dtbuton. Cloud: Cloud doplet ze vay fom a few mcomete to mcomete wth aveage damete n the to µm ange. Cloud doplet concentaton vae fom about cm -3 to cm -3 wth an aveage doplet concentaton of a few hunded cm -3. The lqud wate content of typcal cloud, often abbevated LWC, vae fom appoxmately.5 to 3 g(wate) m -3, wth mot of the obeved value n the. to.3 g(wate) m -3 egon. Cloud doplet ze dtbuton often appoxmated by a modfed gamma dtbuton N( ) N Γ( α) n n α exp( / whee N the total numbe of doplet (cm -3 ); n n the adu that chaacteze the dtbuton ; α n the vaance of the dtbuton, and Γ the gamma functon. n ) [4.3] Table 4. Chaactetc of epeentatve ze dtbuton of ome cloud (fo α ) Cloud type Statu: ove ocean ove land N o (cm -3 ) m (µm) max (µm) e (µm) LWC (g m -3 ) Fa weathe cumulu Matme cumulu Cumulonmbu Altotatu

12 NOTE: Fo many pactcal applcaton, the optcal popete of wate cloud ae paametezed a a functon of the effectve adu and lqud wate content (LWC). The effectve adu defned a 3 π N( ) d e π N( ) d whee N() the doplet ze dtbuton (e.g., n unt m -3 µm - ). [4.3] NOTE: Mean adu: m (α +) n Effectve adu: e (α +3) n The lqud wate content (LWC) defned a 4 3 LWC ρ w V ρw π N( ) d [4.33] 3 Randop Nonphecal patcle: hape depend on ze of a an dop. Randop ze dtbuton often epeented by the Mahall-Palme dtbuton: N( ) N exp( Λ) [4.34] whee N 8x 3 m -3 mm -, but, n geneal, N depend on the an type; Λ elated to the anfall ate, R (n mm/hou) a Λ 4. R -. mm -

13 Ice cytal: Sze ange fom ~5- µm to a few mm Exhbt a lage vaety of hape (called habt): fo ntance, plate - nealy flat hexagon; column - elongated, flat bottom; needle - elongated, ponted bottom; dendte - elongated am (x), nowflake hape. Fgue 4.3 Ice cytal hape a a functon of upeatuaton and tempeatue a clacal vew 3

14 Refactve ndce of wate, ce, and aeool pece. Refactve ndex (o optcal contant), mn κ, the mateal popete of delectc that detemne t adatve popete. In geneal, each mateal ha t own pectal efactve ndex. The magnay pat κ of the efactve ndex detemne the abopton of the wave a t popagate though the medum; the eal pat n of the efactve ndex gve the phae velocty of popagaton. Fgue 4.4 The efactve ndex of wate and ce n the vble and nea-ir. NOTE: wate ha low magnay pat n the vble > neglgble abopton by wate dop n the vble 4

15 Fgue 4.5 The efactve ndex of wate and ce n the IR. 5

16 Fgue 4.6 Clacal plot of the magnay pat of the efactve ndexe of ome aeool mateal a a functon of wavelength (n µm) (Bohen and Huffman, Fg.5.6). NOTE: Man abobng pece n the ola pectum ae black cabon (oot) and hematte (dut), but n the themal IR vaou pece have hgh magnay pat of the efactve ndex. NOTE: Aeool patcle often cont of eveal chemcal pece (called the ntenal mxtue). Thee ae eveal appoache (called mxng ule) to calculate an effectve efactve ndex m e of the ntenally mxed patcle by ung the efactve ndce of the ndvdual pece. Fo ntance, Volume weghted mxng: m e m j f [4.35] j j whee m j the efactve ndex of the j-pece and f j t volume facton. 6

17 5. Scatteng and abopton by aeool and cloud patcle: Me theoy. Me theoy decbe the catteng and abopton of electomagnetc adaton by phecal patcle though olvng the Maxwell equaton. NOTE: Me theoy alo called Loenz-Me theoy o Loenz-Me-Debye theoy. Me theoy outlne Geneal outlne of Me theoy: Key Aumpton: ) Patcle a phee; ) Patcle homogeneou (theefoe t chaactezed by a ngle efactve ndex mn - κ at a gven wavelength); NOTE: Me theoy eque the elatve efactve ndex that the efactve ndex of a patcle dvded by the efactve ndex of a medum. Fo a m about, o one need to know the efactve ndex of the patcle (.e., efactve ndex of the mateal of whch the patcle compoed). NOTE: If a patcle ha complex chemcal compoton uch a ome atmophec aeool, the effectve efactve ndex mut be calculated at a gven wavelength. Me theoy calculate the catteed electomagnetc feld at all pont wthn the patcle (called ntenal feld) and at all pont of the homogeneou medum n whch the patcle embedded. Fo all pactcal applcaton n the atmophee, lght catteng obevaton ae caed out n the fa-feld zone (.e., at the lage dtance fom a patcle). In the fa-feld zone (.e., at the lage dtance R fom a phee), the oluton of the wave equaton can be obtaned a E E l exp( kr + kz) S kr S 4 S 3 E S E l [4.36] 7

18 hee k π/, E l and E ae the paallel and pependcula component of ncdent electcal feld, and El and catteed electcal feld, E ae the paallel and pependcula component of S S 4 ( Θ) ( Θ) S3( Θ) S ( Θ) the ampltude catteng matx (untle) Fo phee: S 3 (Θ) S 4 (Θ), and thu Eq.[6.] gve E E l exp( kr + kz) kr S ( Θ) E S( Θ) E Eq.[4.37] a fundamental equaton of catteed adaton by a phee ncludng polazaton. l [4.37] Me theoy oluton fo the catteng ampltude: S S n + [ a n (coθ) + b (coθ ] ( Θ) n n n ) n n( n + ) n + π [4.38] [ b n (co Θ) + a (co Θ ] ( Θ) n n n ) n n( n + ) whee π n and n ae Me angula functon whee π [4.39] π n(coθ) Pn (coθ) [4.4] n( Θ) d n(coθ) Pn (coθ) [4.4] dθ Pn ae the aocated Legende polynomal. Me theoy alo gve the catteng phae matx P(Θ) that elate the Stoke paamete {I, Q, U and V } of ncdent adaton feld and the Stoke paamete {I, Q, U and V}of catteed adaton: 8

19 I Q σ U 4π V I o Q P U V o o [4.4] whee P P P P P [4.43] P 33 P34 P34 P44 NOTE: In geneal, fo a patcle of any hape, the catteng phae matx cont of 6 ndependent element, but fo a phee th numbe educe to fou. Fo phee: P P and P 44 P 33 Thu fo phee, Eq.[4.4] educe to I P P I o Q σ P P Qo [4.44] U 4π P P U V P34 P33 Vo and each element of the catteng phae matx expeed va the catteng ampltude S (Θ) and S (Θ). P (Θ) P(Θ) the catteng phae functon of a patcle. Fom Me theoy t follow that the extncton co-ecton of a patcle 4 π σ Re[ S( e )] [4.45] k But fo the fowad decton (.e. Θ ) fom Eq.[4.38]-[4.39], we have S ( ) S ( ) (n + )( a n + b n ) n END of Me theoy outlne

20 Effcence (o effcency facto) fo extncton, catteng and abopton ae defned a Q σ π σ π e a e Q Qa σ [4.46] π whee π the aea of a atcle wth adu. Me theoy gve the oluton fo Q e, Q and Q a n tem coeffcent a n and b n (.e., coeffcent n the expeon fo the catteng ampltude S ( ) and S ( )). Q e [4.47] x (n + ) Re[ a + ] n bn n Q [4.48] x (n + )[ a + ] n bn n and the abopton effcency can be calculated a Q a Q Q [4.49] e Fgue 4.7 Example of Q e calculated wth the Me theoy fo eveal efactve ndexe.

21 Some hghlght of Me theoy eult: Extncton effcency v. ze paamete x (aumng NO ABSORPTION): 4 ) mall n Raylegh lmt: Q e x ) laget Q e when patcle and wavelength have mla ze 3) Q e -> n the geometc lmt ( x ) 4) Ocllaton (ee Fg.4.7) fom ntefeence of tanmtted and dffacted wave Peod n x of ntefeence ocllaton depend on the efactve ndex. Abopton educe ntefeence ocllaton and kll pple tuctue. Scatteng and abopton effcence v. ze paamete wth ABSORPTION: A x : Q and, enteng ay ae abobed nde patcle. Smalle magnay pat of the efactve ndex eque lage patcle to fully abob ntenal ay. Scatteng phae functon: fowad peak heght nceae damatcally wth x. Fo ngle patcle numbe of ocllaton n P(Θ) nceae wth x. Fo a ngle phecal patcle, Me theoy gve the extncton, catteng and abopton co-ecton (and effcency facto), the catteng ampltude and phae matx. Integaton ove the patcle ze dtbuton: If patcle have the ze dtbuton N(), the volume extncton, catteng and abopton coeffcent (n unt LENGTH - ) ae detemned a k k e k a σ ( ) N( ) d π Q N( ) d [4.5] e e σ ( ) N( ) d π Q N( ) d [4.5] a ( ) N( ) d σ π Q N( ) d [4.5] a Smla to Eq.[3.], the optcal depth of an aeool laye (between and )

22 (, ) ke, d Mkm, e, d whee k m,e, the ma extncton coeffcent and M the ma of patcle. NOTE: Ma coeffcent volume coeffcent/patcle ma concentaton, M The ngle catteng albedo gve the pecentage of lght whch wll be catteed n a ngle catteed event and t defned a NOTE: No abopton (conevatve catteng): ω No catteng: ω k ω [4.53] ke Scatteng phae functon of patcle wth the ze dtbuton N(): P( Θ) P( Θ, ) σ N( ) d σ N( ) d [4.54] Fgue 4.8 Example of epeentatve catteng phae functon (at a wavelength of.5 µm) fo aeool and cloud patcle. The molecula (Raylegh) alo hown fo compaon.

23 How to calculate optcal chaactetc of an enemble of phecal patcle: Patcle ze Refactve ndex, m() Me theoy Scatteng co ecton, σ Abopton co ecton, σ a Extncton co ecton, σ e Phae functon, P (Θ) (a a functon of patcle ze and wavelength) Integaton ove ze dtbuton N() Scatteng coeffcent, k Abopton coeffcent, k a Extncton coeffcent, k e Phae functon, P (Θ) (a a functon of wavelength) whee Optcal popete of the extenal mxtue (.e., the mxtue of eveal type of patcle) k e k e k k k e, k and k [4.55] a k a k a ae calculated fo each patcle type chaactezed by t patcle ze dtbuton N () and a efactve ndex (o effectve efactve ndex) m. NOTE: Do not um the ngle catteng albedo and catteng phae functon!!! See below Eq.[ ] how to do t. 3

24 How to calculate the effectve optcal popete of an atmophec laye contng of ga and aeool (and cloud): In geneal, an atmophec laye cont of molecule, aeool and/o cloud patcle. Thu, one need to calculate the effectve optcal popete of the laye: Effectve (alo called total) optcal depth: whee M a, and M M A A a, +, + a, +, [4.56] M, ae optcal depth due to abopton by gae and molecula (Raylegh) catteng, epectvely; A A a, and, ae optcal depth due to abopton and catteng by aeool patcle, epectvely. Effectve ngle catteng albedo: ω, Effectve catteng phae functon: Effectve aymmety paamete: M A, +, [4.57] P Θ + P Θ P ( Θ ) [4.58] g M M A A, ( ), ( ) M A, +, g [4.59] A, M, + A A, 6. Remote enng applcaton baed on meauement of dect ola adaton (ee alo Lab 3). Dect adaton a pat of the adaton feld that ha uvved the extncton pang a laye wth optcal depth and t obey the Bee-Bougue-Lambet law (o Extncton law): I d I exp( / µ ) [4.6] whee I the ncdent ntenty at a gven wavelength at the top of a laye and µ a cone of the ncdent zenth angle θ (µ co(θ )). 4

25 Applyng the Extncton law to dect ola adaton: I the ola ntenty at a gven wavelength at the top of the atmophee top θ I o uface I d Thu dect ola adaton eachng the uface I d I exp( / µ ) [4.6] whee the optcal depth on the ente atmophee. NOTE: Optcal depth defned along vetcal coodnate (.e., alttude z) n the atmophee Reteval of aeool optcal depth fom gound-baed unphotomete meauement (lab 3): A unphotomete (a naow-feld of vew adomete that tack the un) meaue the dect ola adaton attenuated by the atmophee: * F d, F, exp( / µ ) [4.6] whee F d, the downwad dect ola flux eachng the uface (.e., the downwad dect ola ntenty ntegated ove the vewng angle of the unphotomete) and * the optcal depth of the ente atmophec column,.e. z top * ( z,) k ( z dz [4.63] top e, ) * Fo cloud-fee atmophec condton, due to attenuaton by aeool, molecula (Raylegh) catteng and gaeou abopton (e.g., O 3 and NO dependng on ). Thu Fom Eq.[4.6] we have * M M a, +, + A [4.64] 5

26 ln( F d µ [4.65] *, ) ln( F, ) / and {ln( F ) ln( F d, )} [4.66] * µ, Thu A M µ {ln( F, ) ln( F d, )} { a, +, M } [4.67] NOTE: To eteve the aeool optcal depth, one need to coect fo Raylegh catteng and gaeou abopton. Othe applcaton: ) If the aeool optcal depth eman contant dung the day, Eq.[4.67] enable to meaue the ola flux F, by plottng meaued F d, v. µ o (called the Langley plot). ln (F o, ) ln ( F d, ) 4 6 m/ µ o ) If the aeool optcal depth known o neglgbly mall, Eq.[6.36] gve the optcal depth due to abopton of gae (ued n the eteval of O 3 and H O column amount). 6

27 Example: AERONET (AEool RObotc NETwok) pogam a fedeaton of gound-baed emote enng aeool netwok etablhed ognally by NASA and CNRS and expanded by many ntenatonal agence, unvete, etc. The pogam povde a long-tem, contnuou and eadly acceble publc data of aeool optcal popete (and wate vapo column amount) fo eeach and valdaton of atellte aeool optcal depth eteval. Fgue 4.9 Cmel Sunphotomete opeated by AERONET. Typcal pectal band (channel) of the Cmel unptomete (cental wavelength): 34 nm, 38 nm, 44nm, 5 nm, 675 nm, 87 nm and nm. Meauement of aeool pectal optcal depth can be ued to compute the Angtom paamete (alo called the Angtom exponent) whch elate to the aeool patcle ze dtbuton: fo meauement of optcal depth and taken at two dffeent wavelength and epectvely, the Angtom exponent α gven by o α [4.68] NOTE: Fo a (pue molecula catteng): α 4 and fo lage patcle (compaed to the wavelength) α 7