18. Atmospheric scattering details

Transcription

1 8. Atmospheric scattering detais See Chandrasekhar for copious detais and aso Goody & Yung Chapters 7 (Mie scattering) and 8. Legendre poynomias are often convenient in scattering probems to expand the phase function. The preferred (my preferred) definition for Legendre poynomias is: () d[ P (cos )] 4 First severa: P P cos P /(3cos ) 3 P /(5cos 3cos ) 3 They form an orthonorma basis set. In order to generate them: ( ) P (cos ) ()cos P P. Phase function expansion is given in genera as: () mp(cos ). Expansion of the phase function is important in radiative transfer modeing. The required number of expansion terms is imited by the number of terms in the radiative transfer expansion itsef (about which more ater). For Rayeigh scattering, m = m = m = / m > = 3 ( cos ) 5 (3cos ). 4 A Rayeigh with depoarization (because of the Raman component, as before) is: 3 (( ) ( )cos ), where is the depoarization factor (=.95 for 4 I H air at 4 nm 9, for unpoarized input (check that for IV pure Rayeigh scattering!) for this phase function, m = m =

2 m 5 m > = Mie scattering Aerosos and couds, especiay. Horriby compicated genera soutions, ots of osciations in phase functions, which average out over size distributions. Detais in Goody and Yung, Chapter 7, and in notes from J. Wang. Note the distinction between absorbing and non-absorbing aerosos: Compex index of refraction, mˆ nˆ in ˆ. (NB back carbon vs. sufates, couds) Transmission exp d E E. i t Since ˆ ˆ / mn, eads to extinction. A typica Mie setup for computation (W. Wiscombe in Disort test code) has 8 Legendre terms in a typica haze and 99 terms in coud. Mie scattering is strongy forward-peaked (tea kette exampe), sometimes with a secondary backward, structured peak (a gory). The Henyey-Greenstein phase function is a common practica Mie phase approximation with nice anaytic properties: g HG(cos, g), where g is the asymmetry parameter. g.6 is 3/ g gcos typica for atmospheric aeroso. This Henyey-Greenstein phase function misses the back scattering peak. This can be treated using the doube Henyey-Greenstein phase function: b (cos, g ) ( b) (cos, g ), where g. HG HG Goody and Yung give a typica atmospheric exampe (maritime m): g.84, g.55, b.974. For HG, m =, m =,, ( ) HG mo m g For the doube HG,, m m bg ( b) g Finay, note the weak waveength-dependence of Mie scattering compared to Rayeigh scattering; it is sometimes.(or some other ow power)

3 For Mie scattering by couds and aerosos, the most common distribution of sizes is ognorma: exp, dn() r c (nr n r ) dr r where is the shape parameter, the n of the standard distribution in width. Shette and Fenn (see references) is a standard source for aeroso information. They describe atmospheric aeroso distributions as either one or the sum of two og-norma distributions. Detais on aerosos to come! -Stream, pane-parae formuation Reca that k m and / scattering is conservative. m Setup for pane-parae atmosphere: (singe scattering abedo). If, the s = norma or zenith ange = cos = optica depth (not optica thickness) μ is defined as cos (θ) where θ is (often) the zenith ange ZA; sometimes the nadir ange (π ZA); and even cos( ZA). We wi use cos cos( ZA). For scattering probems without expicit azimutha dependence, for an arbitrary scattering function, G G( ) d G(, )sindd G(cos )sind G( ) d, a convenient and usefu substitution (which assumes, as is normay the case, that expansion in cos θ is vaid.). Differentia scattering event: I k, for starters) I (just absorption and emission,

4 Id Bd, I B Aready in the basic Schwarzschid form of a radiative d transfer equation (The most basic Schwarzschid form is I J or, expanding τ, d I J ). It is thus sometimes preferabe to keep a dimension: k ds ds I B, so d ds Scattering for ayered atmospheres can be formuated with as the independent variabe (usuay) or z (or P) as the independent variabe (sometimes). is easier mathematicay but tougher when poychromatic (spectra) probems are addressed because ( ) but z z( ). Aso note, however, that when changes rapidy (ike at a coud top), using z or P may require extra effort (ike creating very fine vertica ayers). Add scattering out of beam scattering source term Ids Bkds mi( ) ds (Remember k m. ) Ids B( ) ( ) ds I ( ) ds. Writing B( ) formay acknowedges that the backbody emission has a phase function, even though it is isotropic. Combining source terms (J is the standard symbo for the source): J B( )( ) I ( ). Then, I J. ds We sti need a more compex source (e.g., I might be a penci beam source or a parae bunde, ike I ; the Sun) and recast with d ds: d Id B( )( ) d ( ) (, ). 4 I B( ) ( ) (, ) I J. d 4 Stratified pane-parae atmosphere Goody and Yung.3.3; Chandrasekhar Chapters I and II > (outward, upward) d I (, ) I (, ) e J(, ) e ( )/ ( )/ s s function of reative

5 < (inward, downward) + because < I (, ) I (, ) e J(, ) e / d ( )/ Then, for fuxes (e.g., radiative baance), integrate over d (noting that the BRDF of the surface may introduce -dependence). For soar radiation and negigibe therma source, cacuate I and use abedo: F a a at surface (F = fux); then J F (surface). F Pane parae, direction from boundary inward (or downward) See Chandrasekhar Chapters I and II for detais I J d change in radiative transfer equation since μ= < for the downward direction. Note apparent sign Two-stream probem (up and down) Simpest case: isotropic scattering (Schuster and Schwartzchid, from Chandrasekhar, eading to ater discussions of quadrature*) I(, ) d d (, ) I(, ) d (the factor of ½ is from integrating over azimuth - π and dividing by the isotropic phase function - 4π), or (, ) I(, ) I(, ) d. d Then, defining I and I from sign ( ), I ( I I ) d I ( I I ) d

6 The extra factors of / on the eft come from averaging both extinction and source terms as sincos d / sind / (the mean obiquity of the rays). For conservative scattering ( =, Chandrasekhar, Section ) I I. More reaistic:,. d d *Gaussian quadrature seeks to obtain the best numerica estimate of an integra by picking optima abscissas at which to evauate the function See Chandrasekhar Sections -.