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# Special Topics EAS 8803

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Lectures 14 15 Light scattering and absorption by atmospheric particulates Part 2 Scattering and absorption by individual spherical particles Obiectives l Maxwell equations Wave equation Dielectrical constants of a medium 2 MieDebye theory 3 Optical properties of an ensemble of spherical particles Required Reading L02 52 332 AdditionalAdvanced Reading Bohren GF and DR Huffmn Absorption and scatteiing of light by small particles John WileyampSons 1983 Mie theory deiivation is given on pp82ll4 A 1 Maxwell quot Wave emmtinn Dielectrical of a medium Maxwell equations connect the ve basic quantities the electric vector E magnetic vector H magnetic induction 3 electric displacement I3 and electric current density in cgs system M124 00 t c VXE1 141 c 01 V05 47z39p vo 0 where c is a constant wave velocity and p is the electric charge density To allow a unique determination of the electromagnetic eld vectors the Maxwell equations must be supplemented by relations which describe the behavior of substances under the in uence of electromagnetic eld They are jaE 5 gi y Mm where 039 is called the speci c conductivity 8 is called the dielectrical constant or the permittivity and u is called the magnetic permeability 7 139 0n the value of a39 the L are divided into conductors Oquot i 0 ie 039 is NOT negligibly small for instance metals dielectrics or insulators Oquot 0 ie 039 is negligibly small for instance air aerosol and cloud particulates Let consider the propagation of EM waves in a medium which is a uniform so that a has the same value at all points b isotropic so that s is independent of the direction of propagation c nonconducting dielectric so that c 0 and therefore j 0 d free from charge so that p 0 With these assumptions the Maxwell equations reduce to vXH 2 c at VXE 7 uai c at 143 vEo VIH0 Eliminating E and H in the first two equations in 143 and using the vector theorem we have 2 WE 5 if 144 6 via i 6 2 612 The above equation are standard equations of wave motion for a wave propagating with a velocity v 145 V 5 where c is the speed of light in vacuum NOTE for vacuum p 1 and 8 1 in cgs units but in SI system no and so are constants such that c 1 smug For most substances including the air u is unity Thus the electrical properties of a medium is characterized by the dielectrical constant 8 Refractive index or optical constants of a medium is de ned as m Z 146 assuming that u1 NOTE Strictly speaking a in Eql46 is the relative permittivity of medium here it is relative to vacuum Refractive index lt The refractive index mmr imi is commonly expressed as a complex number The nonzero imaginary part m of the refractive index is responsible for absorption of the wave as it propagates through the medium whereas the real part m of the refractive index relates to the velocity of propagation of the EM wave The refractive index is a strong function of the wavelengths Each substance has a specific spectrum of the refractive index see figures 5657 Lecture 5 Particles of different sizes shapes and indices of refraction will have different scattering and absorbing properties Aerosol particles often consist of several chemical species called the internal mixture There are several approaches called mixing rules to calculate the effective refractive index n1e of the internally mixed particles using the refractive indices of the individual species see Lecture 5 gt Scattering domains Rayleigh scattering 21ml ltlt1 and m is arbitrary applies to scattering by molecules and small aerosol particles RayleighGans scattering 277 m 7 1 ltlt land lm 7 1 ltlt 1 not useful for atmospheric application MieDebye scattering 21lrl and m are both arbitrary but for spheres only applies to scattering by aerosol and cloud particles Geometric optics 21m gtgt1 and m is real applies to scattering by large cloud droplets E E 7 S g E e gt E 8 E 2 3 gt Z l E gt I I I I I I Ls 39 1cm Q9 Hail a Qquot I mm 320 0 4130 Raindrops 100le 36 Drizzle m x x x 2 quot L3 D P x x g lOIIm Cloud droplets in E 1 701 as 1 Dust g 1 I XI 5amp6 xx Smoke 393 1 QK 1 Haze 0 lim xx Q5692 Q96 9 i go Aiiken Nuclei L36 10nm I vquot 2 W x x 2 1 nm xquot Air Molecules I I I I I 01 um llim loum 100 lim 1 mm 1cm 10cm Wavelength Figure 141 Relationship between particle size radiation wavelength and scattering behavior for atmospheric particles Diagonal dashed lines represent rough boundaries between scattering regimes 2 Mie Debye theory NOTE MieDebye theory is often called Mie theory or Lorentz Mie theory Mie th ear outline Assumptions i Particle is a sphere of radius a ii Particle is homogeneous therefore it is characterized by a single refractive index mmr imi at a given wavelength NOTE Mie theory requires the relative refractive index refractive index of a particlerefractive index of a medium But for air In is about 1 so one needs to know the refractive index of the particle ie refractive index of the material of which the particle is composed NOTE If a particle has complex chemical composition the effective refractive index must be calculated at a given wavelength Strategy 1 Seek a solution ofa vector wave equation Eql44 for E and H W 8 6 Z V 2E 0 27 with the boundary condition that the tangential component of E and H be continuous across the spherical surface of a particle Assumption on the spherical surface of a particle allows solving the vector equation anallgticalllg 2 Rewrite the wave equation in spherical coordinates and express electric field inside and outside sphere in a vector spherical harmonic expansions NOTE Mie theory calculates the electromagnetic field at all points in the particle called internal field and at all points of the homogeneous medium in which the particle is embedded For all practical applications in the atmosphere light scattering observations are carried out in the farfield zone ie at the large distances from a sphere 3 Apply boundary conditions 7 match transverse fields at sphere surface to obtain scattered spherical wave Mie coef cients an and bn which don t depend on the angles but depend on size parameter x 27ml a is the radius of the particle and variable y x m m is refractive index of the particle 4 Use series involving an and bn to obtain extinction and scattering ef ciencies Q8 and Q5 5 Use series in Mie angular functions n and 1 to obtain scattering amplitude functions S1 and S2 from which the scattering phase function is derived NOTE Full derivation of Mie theory are given in L02 section 52 and BohrenampHuffman 1983 pp82l 14 M ie scattering amplitudes also called scattering functions derived from Mie theory are see Eqs5278 in L02 S1 26575 cos bnrncos 9 529 2bn ncos anrncos 9 147 where Mie coefficients an and bn are see Eqs5274 in L02 11ytxe mlIyz x b x y mzI ylxe 11 yl x 143 away 11 mi x e m 11 mi x m 11 mi x e 11 mi x here the prime denotes differentiation x 27111 and y X m Wm lgjwzm and 5amp0 gHJp where Jn1Zp is the halfintegralorder spherical Bessel function and HZ is the halfintegralorder Hankel function of the second kind and 1t and 1 are the Mie angular functions it cos cos 239 cos 9 P cos 149 where P are the associated Legendre polynomials see Appendix E In the far eld zone ie at the large distances r from a sphere Mie theory gives the solution of the vector wave equation can be obtained as E S S E 1 eXp zkrzlltz 2 3 1 1410 E z39kr S4 S1 E Eq14 10 is a fundamental equation of scattered radiation including polarization in the far eld 526 536 546 516 is the amplitude scattering matrix unitless For spheres S3 S4 0 Thus for spheres Eq14 10 reduces to E W 32 0 1411 E ikr 0 S1 I where eXp ikz is the incident plane wave and m is the outgoing scattered 739 wave Fundamental extinction formula or optical theorem gives the extinction cross section of a particle 2 47239 039 k ZReS1720 1412 But for the forward direction ie 00 from Eq14 7 we have 5100SZOOZ2nlan 13 1413 n1 Thus extinction cross section is related to scattering in forward direction gt m or m factors for 39 39 scattering and absorption are de ned as 7 5 e 5 6 Q2 7 2 Q5 2 Q 2 1414 7ra 7ra 7ra where m2 is the particle area projected onto the plane perpendicular to the incident beam Mie ef ciency factors are derived from the Mie scattering amplitude Q2 izi 2nlRe anbn 1415 x n1 Q5 2 i 2nlan2bn2 1416 and the absorption ef ciency can be calculated as 1417 Extinction Efficiency Absorption Efficiency 4 I I I I I I I I 2 I I I I I I I I I m 133 m 133 000M 35 mi330ii m i3300 ii m133101 7777 n quot m3301 15 m1331oi r r 7 Wmmmtmmmmmwwwwww r fi39wir I I I I I I 0 5 10 15 20 25 30 35 4O 45 50 O 5 10 15 20 25 30 35 40 45 50 Size parameter errLrA Size parameter ernrA 0 I I I I I I I I I Figure 142 Examples of Q8 and Q8 calculated with Mie theory for several refractive indexes gt Scattering phase matrix Recall de nition of Stokes parameters see Lecture 13 which uniquely characterize the electromagnetic waves Let Io Q0 U0 and V0 be the Stokes parameters of incident eld and I Q U and V be the Stokes parameters of scattered radiation 1 10 Q 03 P Q 1418 U 47 2 U0 V V a where P is the scattering phase matrix P11 P12 0 0 P P 0 0 P 12 22 1419 0 0 1333 P34 0 0 PM PM where each element depends on the scattering angle lr2 is from solid angle For spheres P22 P11 and P 44 P33 NOTE In general for a particle of any shape the scattering phase matrix consists of 16 independent elements but for a sphere this number reduces to four Thus for spheres Eql418 reduces to 1 P11 P12 0 Q 03 P12 P11 Q0 1420 U 47 quot2 P33 7 P34 U0 V P34 P33 Va where each element of the scattering phase matrix is expressed Via the scattering amplitudes S1 and S2 4 7239 1 1 P11 WS1S1 SZSZ P12 4 SZS S1SI 143921 ZkZUS 4 a 260 ssss 4 7 PM 260 m e ss Pne P6 is the scattering phase function de ned in Lecture 13 Phase function W U 0 20 40 60 80 100 120 140 160 180 Scattering angle Figure 143 Examples of scattenng phase functh calculated wlth Mre Lheoxy for several srze parameter for nonabsorbmg spheres 11 IA Some 39 39 0f Mie scatterino results 0 Extinction efficiency vs size parameter x assuming NO ABSORPTION 1 small in Rayleigh limit Q6 0C x4 2 largest Qe when particles and wavelength have similar size 3 Q6 gt 2 in the geometric limit x a co 4 Oscillations see Fig 143 from interference of transmitted and diffracted waves 0 Period in x of interference oscillations depends on the refractive index Absorption reduces interference oscillations and kills ripple structure 0 Scattering and absorption efficiencies vs size parameter with ABSORPTION As x a co Q5 1 and entering rays are absorbed inside particle Smaller imaginary part of the refractive index requires larger particle to fully absorb internal rays 0 Scattering phase function forward peak height increases dramatically with x For single particles 7 number of oscillations in P increases with x NOTE Several Mie codes are freely available to the scientific community see httpatolucsdedup atauscatlibscatterlibhtm How a Mie code works 1 Compute coefficients an and bquot see Eq 14 8 for nl N from size parameterx and index of refraction m uses recursion relations for the spherical Bessel functions Estimate of the total number N x 4x132 2 Compute Q6 and QS from an and bquot 3 Compute scattering amplitude functions S1 and Sz at desired scattering angles from an and bquot and Mie angular functions 7239quot O and rquot O n and rquot from recursion Compute phase matrix elements P11 P12 P33 and P34 from S1 and Sz 4 If optical properties of an ensemble of particles are required one needs to perform an integration over the size distribution function 3 Optical m onerties of an of snherical narticles Mie theory gives the extinction scattering and absorption crosssections and ef ciencies and the scattering phase matrix of a single spherical particle NOTE Recall Lecture 5 where the particle size distributions were introduced for atmospheric aerosols and clouds If the particles characterized by a size distribution Nr the volume extinction scattering and absorption coefficients in units LENGTH39I are calculated as 6 2 39T0392rNrdr rTasrNrdr 1422 Taxomrwr where 6 is the corresponding cross section of a particle of radius r and Nr is the particle size distribution eg in units m393um391 Single scattering albedo unitless if de ned as 1423 The single scattering albedo gives the percentage of light which will be scattered in a single scattered event Scattering phase function is no 5T7 5155 SZS Nrdr 1424 P aSNrdr or P 39m f 1425 Asymmetpy parameter is de ned as the first moment of the scattering phase function g 1Pcos cos dcos 1426 g 0 for equal forward and backward scattering g l for totally forward scattering The Henyey Gr eenstein scattering phase function is a model phase function which is often used in radiative transfer calculations legz 1427 P HG 1g272gcos 32 where g is the asymmetry parameter Optical properties of cloud particles 0 For many practical applications the optical properties of water clouds are parameterized as a function of the effective radius and liquid water content LWC The effective radius is defined as 7 I r 3 N r dr 1428 INN rdr 2 where Nr is the particle size distribution e g in units m393um391 The liquid water content LWC was defined in Lecture 5 see eq510 LWC pWV gpwjmwwr 1429 Using that the extinction coefficient of cloud droplets is jaltrN rm 9 er rm and that Q m 2 for water droplets at solar wavelengths we have 3LWC 1430 6 2 new a Single Scatter Albedo 7 Water vs ice 1 Walel l 0 Mn lL el e 20 11m 7 U I I 1 111 Wavelength 7 um C Single Scatter Albedo Cloud Droplets 1 A a u I l 1 0 Wavelength um b Single Scatter Co AIIedo 7 Water vs ice 1 uuu1 r u mum r Walel l e 20 wn iliel e zuI W 1e us 0 1 1 5 2 Wavelength 1 lvm d Single Scatter C0Albedo 7 Cloud Droplets 1 o om o 0001 1e 05 u Wavelength ll um Figure 1414 Single scattering albedo or coalbedo 1 no as a function ofWavelength for Water and ice spheres of varying sizes Lecture 25 Radiation d in climate and energy balance models Obiectives l A hierarchy of the climate models 2 Examples of simple energy balance models 3 Radiation in climate models Required reading L02 85 861 Additional reading The NCAR Community Climate Model httpwwwcgducareducmsccm3 1 A hierarchy of climate models 0 Climate models can be classi ed by their dimensions Zero Dimensional Models 0D consider the Earth as a whole no change by latitude longitude or height One Dimensional Models lD allow for variation in one direction only e g resolve the Earth into latitudinal zones or by height above the surface of the Earth Two Dimensional Models 2D allow for variation in two directions at once eg by latitude and by height Three Dimensional Models 3D allow for variation in three directions at once ie divide the earthatmosphere system into domains each domain having its own independent set of values for each of the climate parameters used in the model 0 Climate models can be classi ed by the basic physical processes included into the consideration Energy Balance Models 0D or lD models eg allow to change the albedo by latitude calculate a balance between the incoming and outgoing radiation of the planet Radiative Convective Models see lecture 24 1D models to model the temperature pro le the atmosphere by considering radiative and convective energy transport up through the atmosphere General Circulation Climate Models 2D longitudeaveraged or 3D climate models solve a series of equations and have the potential to model the atmosphere very closely 2 Examples of simple energy balance models Recall Lecture 2 Planetary radiative equilibrium TOA outgoing radiation TOA incoming radiation over the entire planet and long time interval eg a year Let s estimate the effective temperature assuming that the Earth is in the radiative equilibrium The sun emits F5 62XlO397 Wm2 a blackbody with about T 5800K From the energy conservation law we have Fs 4111 Fso 41cD02 where Rs is the radius of the sun 696XlO5 km F90 is the solar ux reaching the top of the atmosphere called the solar constant about 1368 Wmz at the average distance ofthe Earth from the sun D0 15XlO8 km Thus we have Fso Fs Rsz D02 If the instantaneous distance from the Earth to sun is D then the total sun energy ux F0 reaching the Earth is F0 Fso DoD2 The total sun energy intercepted by the cross section of the Earth is F50 39ltRe2 where Re is the radius of the Earth This energy is spread uniformly over the entire planet with surface area 41tRez Thus the amount of received energy per unit surface becomes F90 KR2 mm2 Fso 4 Therefore the total energy Qin in Wmz absorbed by the earthatmosphere system is Qin1 39 7 F90 4 where I7 is the spherical or global albedo see Lecture 18 Spherical albedo of the earth is about 03 Assuming that the Earth is a blackbody with temperature Te we have 0 E cg Te where 039 is the StefanBoltzmann constant From the balance of incoming and outgoing energy the effective temperature of the Earth is Qin Qout Fso 1 7 4O39B Te Te Fso1 74cyB Te 255 K 18 C is very low T8 is much lower than the global average surface temperature about 288 K Why Because we didn t include the greenhouse effect and ignore the temperature structure Table 251 Effective temperatures of some planets in the radiative equilibrium Relative distance to Planet the sun with respect Global albedo TeK How can we measure the greenhouse effect Upwelling ux at the surface ET 2 0T Upwelling ux FTTOA at the top of the atmosphere TOA from satellite observations The difference between the upwelling uxes at the surface and TOA gives a measure of greenhouse effect G G 2 0T FTA 251 NOTE G is the amount of heat eg measured in Watts per unit area of the Earth What is a reasonable estimate of the greenhouse effect FT 235 Wmz Ts 288 K G 390235 155 Wm2 oTJ4 390 Wm2 Sim le model 0 reenhouse e ect 1 sin le la er ra ener balance model Let s include the atmosphere assuming that it emits absorbs as a gray body Assume that the atmosphere does not absorb solar radiation all is absorbed at the surface F TOA balance 701 7 l an SGT 252 F Surface balance 701 7 SGT O39TS4 253 where 6 is the StefanBoltzmann constant and s is the emissivity of the atmosphere F 1 5 F 254 202 e for s 06 gt Ts 278 K NOTE increasing for 8 increases Ts This is socalled runaway greenhouse effect warmer Ts gt more evaporation gt more water vapor gt higher emissivity gtwarmer Ts Why Because we assume that atmosphere is a gray body Sim le model 0 reenh ouse e ect 2 sin le black la er with s ectral window ener balance model A more realistic way to deal with partial longwave transparency of the atmosphere is to assume that a fraction of the spectrum is clear Assume that l fzaT 4 J BV Td V is the fraction of LW spectrum which is completely transparent and the remainder of the LW spectrum is black Surface is black and n0 SW absorption by the atmosphere TOA balance 1 7 fO39TS4 l f039Ta4 255 F Surface balance 701 7 l f039Ta4 O39TS4 256 Thus we can express Ts and Ta via Te T 2 TNT and T rl4T S E a E 257 1f 1f Limits N0 window f0 gt T5 2147 and Ta Te A11 window f1gt T5 T8 and Ta TE 2 4 Earth fis about 03 gt Ts 284 K and Ta 239 K H ow to make the model more realistic Tropics radiation excess North poles and high latitudes radiation de cit 3 must be poleward transport of energy Onedimensional atitude ener balance model Budyko 1969 Sellers 1969 Cess 1976 Atmosphere is only implicit TOA outgoing longwave ux is parameterized as a function of Ts Budyko s parameterization is based on monthly mean atmospheric temperature and humidity pro les and cloud cover observed at 260 stations FLWx a1 b1TSx a2 b2Tx77 258 where ai and b are the empirical constants based on statistical tting X sin 1 and I is latitude If cloud cover is taken constant of 05 then FLWxl55Wm2KTSx 212Wm2 259 NOTE The approximation for linear relation between OLR and the surface temperature may be argued from the fact that the temperature pro les have more or less the same shape at all latitudes and that OLR which depend on temperatures at all levels may be expressed as a function of surface temperature Annual mean TOA solar insolation t well with Sx F0 4l 0482P2 x 2510 P2X 3X2 l 2 is the second Legendre polynomial Thus energy balance equilibrium ST5t0 with diffuse transport D xl x2FLW Sxl rx 2511 where D is diffusion coef cient for energy transport and rX is albedo 4U quotanquot 39li 39n Infinite Diffusive Transport TUI l 860 l l I Figure 251 Zonally average surface temperature K as a function of the sine of the latitude u u is same as X in 251 1 observed and for cases of no horizontal heat transport and in nite horizontal heat transport North et al 1981 NOTE with no meridional transport the poles are way too cold 3 Radiation in climate models The stateafar climate models coupled oceanatmosphereland biosphere 3D global circulation models quotcoming Solar Emmy Outgoing Heat Energy Transition tram Solid ta Vapor Cirrus Clouds Evapor live Shams 01 andEHeatEne y Atmosphere r9 quothinges Cumulus VCI OI I dS now Cover Freqipka on Va Evap nrguen tm Vph39erlc Model Layers Figure 252 Schematic representation of processes in the Community Climate System Model CCSM NCAR NOTE A number of climate models has been developed Each climate model has its own radiation block gt problem modeldependent differences in radiative elds Lecture 9 Ocean color Obiectiver 1 Re ection from surfaces 2 Principles of ocean color characterization Rem39 ed re ng39 3 pp 17718063 651 Additionaladvanced new MODIS ocean color httpoceancolorgsfcnasagov Gordon HR Atmospheric correction of ocean color imagery in the Earth Observing System eraJ Geophyr Res 102 1708117106 1997 1 Re ection from surfaces Bidirectional reflectance distribution function BRDF is introduced to characterize the angular dependence in the surface re ection and defined as the ratio of the re ected intensity radiance to the radiation ux irradiance in the incident beam fdliwpw 91 I Mazw where M cos 9 and e is the incident zenith angle q is the incident azimuthal angle RM 1 MM and cos 9 and e is the viewing zenith angle q is the viewing azimuthal angle 2 Vi mg diam m Two extreme gages at the surtace re ection specular reflectance and diffuse reflectance Specular reflectance is the re ectance from a perfectly smooth surface eg a mirror Angle of incidence Angle of re ectance 0 Re ection is generally specular when the quotroughnessquot ofthe surface is smaller than the Wavelength used In the solar spectrum about 04 to 2 urn re ection is therefore specular on smooth surfaces such as polished metal still Water or mirrors 0 Practically all real surfaces are not smooth and the smface re ection depends on the incident angle and the angle of re ection Re ectance from such surfaces is referred to as diffuse reflectance Indusquot Itgm specularly madequot gm ditl39uscly re ected light re e med light medium medium mm surface I t specularly termed object I t amuse t light I t tmnsm led light K 39 1 4quot 39 39 Inmhminn r 1 thin A smface called the Lambert surface ifit obeys the Lambert s Law Lambert s Law of dif ise re ection the dif isely re ected light is isotropic and unpolarized ie natural light independently of the state ofpola1ization and the angle of the incidence light Re ection from the Lambertian surface is isotropic Incident ray summed gm For a Lambertian surface Rpz lR 92 where RL is the Lambert reflectance also called surface albedo which in general depends on the wavelength Examples of the spectral surface reflectance i e surface reflection as a function of wavelength leIeclance 96 0 075111111 sssss naveeng 1111 Figure 91 Surface albedo of snow composed of crystals of d0075 mm and d05 mm NOTE Each surface type has a specific spectral fingerprint the surface re ection has a speci c dependence of the wavelength This plays a central role in the remote sensing of planetary surfaces 2 Principles of ocean color characterization Ocean color sensors Scienti c Motivation Remote sensing of ocean color provides information on the abundance of phytoplankton and the concentration of dissolved and particulate material in surface ocean waters Importance o biological productivity in the oceans the oceans take up about 13 of C02 two major mechanism solubility pump and biological pump the latter is controlled by phytoplankton biomass marine optical properties the interaction of Winds and currents With ocean biology see Figure 92 effects of human activities on the oceanic environment MidAtlantic Ocean Current 08May2000 r 433m 310N I r 731W lXW Figure 92 Surface layer dri red arrows derived from MODIS and SeaWiFS chlorophylla concentration data over the midNorth Atlantic Ocean Liu et al E08 83 2002 ncem water Pnnclgles afacenn calm remems me epeem wntaelenwng quotmnees Ire menslred Need rm eeume ntrna heme e neck Wntaelenvlng quotmnees cnn be 5 law 5 n few paced enhe TOA tapeafethee Ram me ntrnasphere m me TOA quotmnees mmmm eweswmnnon m rugm Mommy mm Imam msw m u z 5 n lt z A quotpom l s T mom lY nm o 2 alum 2a a 5 Moms rams EaumLzm ummcz Figure 93 MODIS wnvelmg39h bmds ma me wnta Ienvnng rndnnce m high ma new duemphyu Wnters wmwm me ntmasphmc sgnnl newer curves ma Wm me ntrnasphenc 59m uppa curves SENSORS CZCS Coastal Zone Color Scanner own on the NIMBUS7 satellite data for 1978 1986 SeaWiFS SeaViewing Wide FieldofView Scanner launched onboard OrbViewZ satellite data from 1997 MODIS Moderate 39 quot Imaging quotr J39 launched on Terra satellite data from 1999 Table 91 MODIS SeaWiFS and CZCS channels and their central wavelengths used for ocean color retrievals MODIS Sea Ocean color retrieval algorithm retrieval of the chlorophyll concentration CZCS SeaWiFS and MODIS algorithms use the normalized waterleaving radiance Iw N defined as 1M 1 WM 93 where T 7 is the diffuse transmittance see Lecture 8 eq89 W 0 is the radiance backscattered out of the water 0 The normalized waterleaving radiance is approximately the radiance that would eXit the ocean in the absence of the atmosphere with the sun in the zenith Lecture 22 Methods for solving the radiative transfer eguation with multiple WW inhomogeneous clouds Obiectives 1 Monte Carlo method 2 Examples of radiative transfer techniques for inhomogeneous clouds 1D with a cloud fraction independent column approximation ICA and SHDOM Required reading L02 67 AdditionaVAdvanced reading The International Intercomparison of 3D Radiation Codes I3RC httpi3rcgsfcnasagov Cahalan R F et al 2005 The International Intercomparison of 3D Radiation Codes I3RC Bringing together the most advanced radiative transfer tools for cloudy atmospheres Bull Amer Meteor Soc 86 9 12751293 Marshak A and ADavis Eds 3D Radiative Transfer in Cloudy Atmospheres SpringerVerlag Berlin 2004 1 Monte Carlo method The absorption and scattering processes in the atmosphere can be considered as stochastic processes Recall that energy of one photon is hcA where h 6626XlO3934 J s Solar ux at the top ofthe atmosphere at 550 nm 255XlO15 photons cm392 s391 Thus the radiative field can be predicted by statistical analysis of traveling photons The scattering phase function can be interpreted as a probability function for the redistribution of photons in different directions The single scattering albedo 00 can be interpreted as the probability that a photon will be scattered given an extinction event NOTE 1 no is called co albedo and can be considered as the probability of absorption per extinction event The concept of the Monte Carlo method is to simulate photon propagation in an optically effective medium as a random process Generation of random numbers Using a random number generator a numerical algorithm the random numbers rn between 0 and l with a probability distribution function PDF can be generated Using this rn we can generate another set of random numbers as rx lnrn with PDF exp rx and rx between 0 and infinity Let s consider a homogeneous medium characterized by the extinction coefficient ext single scattering albedo 00 and phase function Pu u39 Monte Carlo method 39 39 the trajectories of 39 139 391 39 photons accordin to the following scheme 1 Determine starting position x0 and direction pup of a photon 2 Generate a photon path length using random numbers rx x rx 3 Calculate a new photon position x0x called the event point or collision point 4 Analyze what can happen with the photon at this event point by generating the random number rn and comparing it with 00 if rn gt no gt the photon is absorbed gt go to l for a new photon if rn lt coo gt the photon is scattered gt go to 5 5 Find a new direction for the scattered photon using the phase function to calculate the cumulative probability function to relate the scattering angle to a random number 6 Then repeat starting with 3 until the all photons are analyzed Monte Carlo requires about 105 7109 photons to produce statistically reliable results Backward Monte Carlo method starts With the photon at the point of interest and traces back to the source Let s consider the inhomogeneous atmosphere We can split it into the homogeneous grids Point of interaction or event point rx Z mp 3m x yJ step Where step stepsize in each grid 1 13m is called free pathlength I exper expe mx expex I 221 n omo cueous clouds e transfer technl us to yz has llmlted appllcauons m E Error J It assumes 1D assumes noninfernc ng homogeneous pixels pixels 5 100 m 1 km 10 km 100 km on m PIer Size Fimlre 22 I cloudy condroons Wrscombe at al mm on o cloud How to quotat the The slmplest method introduce a cloud fraction rd fclIS commonly reported from meteorologlcal observauorrs I fclIcl 1 01le h I cloud and of clear sky But are problem ls that f obxerved f4 radzazzve Anomer problem ls cloud overlap Independent Column Approximation 1 CA2 ICA is computational ef cient technique to calculate the radiative transfer accounting for the cloud inhomogeneity A cloudis subdividedinto columns planeparallel radiative transfer is applied to each column and the overall radiative transfer effect is the summation from the individual columns Thus ICA calculates the domainaveraged radiative properties ICA requires the probability distribution of optical depth PDF I in the cloudy part of the scene instead of just the mean optical depth ICA concept is well suitable for GCM models but does not work well in the remote sensing of cloud properties Modi cation of ICA NICA nonlocal independent column approximation has been proposedto account for radiative smoothing effect ie the tendency of horizontal photon transport to smooth the radiative eld predicted by ICA 00 300 600 900 HIM 0 an bepm Independent pixel approximation 3D Simulation mm I 7 l M rquot quot 39 aaudresscn se i l clondlesscnso M H quotg 5 i Mi l H i E s NH 500 l l 500 a 5 H l l E r Wu 3 l l E l l 400 ll Mm g 3 l r i r l 2 quot l l g Ll quot1 H c 700 quot oa quot I u l J 4 b D I 1 4 6 x km x km Figure 222 Solar flux 02pm 4pm at the Earth39s surface calculated with ICA and Monte Carlo for the cloud field shown on the upper panels The sun is shining from the left solar zenith angle is 30 The right image shows a cross section along the dotted line Mayer et al NOTE The 3D shadows do not appear below the clouds but of course offset into the direction of the direct solar beam The 3D radiation field oumide the cloud shadows is enhanced compared to the onedimensional approximation The 3D ux inside the cloud shadows is considerably enhanced compared to the independent pixel approximation by more than a factor of 2 This is caused by sidewards scattering of radiation under the cloud Barker et al Assessing lD Atmospheric Solar Radiative Transfer Models Interpretation and Handling of Unresolved Clouds Journal of Climate vol 16 Issue 16 pp26762699 2003 The Monte Carlo Independent Column Approximation McICA combines IAC and Monte Carlo incorporated into the ECMWF forecasting model Lecture 3 Basic Concepts of extinction scattering plus absorption and emission Schwarzschild s eguation Obiectives 1 Basic introduction to electromagnetic eld Definitions dual nature of electromagnetic quantities The BeerBouguerJ ambert law radiation electromagnetic spectrum 2 Basic radiometric quantities energy intensity and flux 3 The BeerBouguerLambert law Concepts of extinction scattering absorption and emission Optical depth 4 Simple aspects of radiative transfer Schwarzschild s radiative transfer equation Required rem L02 11 14 1 Basic introduction to electromagnetic eld Electromagnetic radiation is a form of transmitted energy Electromagnetic radiation it 39 39 mm mm u cumin planes mutually perpendicular to each other and to the direction of propagation through L39L39 c space 39 iau iatiuu 39 nature it in I I particulate properties Electric eld component Direction of propagation Magneticfield kwgt component ave ength A Figure 31 Electromagnetic radiation as a traveling Wave Wave nature of radiation radiation can be thought of as a traveling wave characterized by the wavelength or frequency or wavenumber and speed NOTE speed oflight in a vacuum C 29979XlO8 ms 5 300XlO8 ms Wavelength 3 is the distance between two consecutive peaks or troughs in a wave Frequency 17 is de ned as the number of waves cycles per second that pass a given point in space Wavenumber V is de ned as a count of the number of wave crests or troughs in a given unit of length Relation between 7 V and 17 v 170 17 31 NOTE The frequency is a more fundamental quantity than the wavelength Wavelength units LENGTH Angstrom A l A leO3910 m Nanometer nm 1 nmlX10399 m Micrometer um 1 pm leO396 m Frequency units unit cycles per second Us or s39l is called Hertz abbreviated Hz Wavenumber units LENGTH391 often in cm39l 0 As a transverse wave EM radiation can be polarized Polarization is the distribution of the electric field in the plane normal to propagation direction Particulate nature of radiation Radiation can be also described in terms of particles of energy called photons The energy of a photon is given by the expression Ephumn h 17hc2hcv where h is Plank s constant h 66256XlO3934 J s NOTE Plank s constant h is very small 0 Eq 32 relates energy of each photon of the radiation to the electromagnetic wave characteristics 17 v or 7 gt Sgectrum nfalectrnmametic radiatinn Is he membuuen ufelectxumagnetn radiatan accurdmgtu energy Dr equIveIenuy my accurdmg tu me wavelength Wavenumber Dr frequ mxmxmm WWW wIeIeI5IIm4gt Nr 10 ID 7 If W In In I I I I I I 1 I I I am hm s i 1 my I xIry Imam Inlmml I Mmolwa I mom I I I Inm m mquot mquot w39 m 0 M 10 lt7 FINIqu Lf we 5m 5m 7m 1mm VIxiblzlvsm Figurei39zl The electxungnenc spectrum regjnn III 2 Basic radiometric Quantities Flux and intensity are the two measures of the strength of an electromagnetic eld that are central to most problems in atmospheric sciences Intensi gor radiance is defined as radiant energy in a given direction per unit time per unit wavelength or frequency range per unit solid angle per unit area perpendicular to the given direction see gure 33 1 dE 33 cos6detdAd IA is called the monochromatic intensity Monochromatic does not mean at a single wavelengths 7 but in a very narrow in nitesimal range of wavelength A centered at 7 NOTE same name intensity speci c intensity radiance UNITS from Eq33 J sec391 sr391 m392 um39l W sr391 m392 um39l Figure 33 Illustration of differential solid angle in spherical coordinates Solid angle is the angle subtended at the center of a sphere by an area on its surface numerically equal to the square of the radius a Q r 2 UNITS of a solid angle steradian sr A differential solid angle can be expressed as do 9 d9 r2 s1n 6d6d using that a differential area is dA r 619 r sin9 do EXAMPLE Solid angle of a unit sphere 411 EXAMPLE What is the solid angle of the Sun from the Earth if the distance from the Sun from the Earth is d15XlO8 km and Sun s radius is Rs 696XlO5 km 2 676x105sr Q Properties of intensity In general intensity is a function of the coordinates 7 direction 52 wavelength or frequency and time Thus it depends on seven independent variables three in space two in angle one in wavelength or frequency and one in time lt Intensity as a function of position and direction gives a complete description of the electromagnetic field lt If intensity does not depend on the direction the electromagnetic field is said to be isotropic If intensity does not depend on position the field is said to be homogeneous Adi UNITS from Eq 34 J sec391 m39Z um39l W m392 um39l From Eqs 3334 FA Ill cos8dQ 35 0 Thus monochromatic ux is the integration of normal component of monochromatic intensity over some solid angle 39 unwelling unwarrh 39 ux on a horizontal plane is the integration of normal component of monochromatic intensity over the all solid angles in the upper hemisphere FJ IRQWQ 36 Zir Eq 36 in spherical coordinates gives z ZmrZ 7r 1 Fl I 128wcos8sin8d8dw I II Md dw 37 I I I I Where u code uux i 9 integration over the lower hemisphere l Zr yr 27 7 F 7 inltawcosltasinltadadz2 7 j Vimwade I nZ I I 27 I ll wmwd dw 38 Monochromatic net ux is the integration of normal component of monochromatic intensity over the all solid angles over 411 Net ux for a horizontal plane is the difference in upwelling and downwelling hemispherical uxes 2 1 F Fj J lemmaaw 39 071 Actinic ux is the total spectral energy at point used in photochemistry Em IMQMQ 310 47 SQectral integration Radiative quantities may be spectrally integrated For example the downwellz39ng shortwave ux is l 40W F IFAdl 311 Olym Intensities and uxes may be per wavelength or per wavenumber Since intensity across a spectral interval must be the same we have Add V61 V and thus a dv I 11 11V1 2IMZ 312 v EXAMPLE Convert between radiance in per wavelength to radiance per wavenumber units at 7 10 um Given I 99 W m392 sr39lum39l What is IV IV 99 w m392 sr391 pm39l 10 um 10393 cm 0099 w m392 sr391cm391391 and emission o Extinction and emission are two main types of the interactions between an electromagnetic radiation eld and a medium eg the atmosphere General de nition Extinction is a process that decreases the radiant intensity while emission increases it NOTE same name extinction attenuation Radiation is emitted by all bodies that have a temperature above absolute zero 0 K often referred to as thermal emission 0 Extinction is due to absorption and scattering Absorption is a process that removes the radiant energy from an electromagnetic eld and transfers it to other forms of energy Scattering is a process that does not remove energy from the radiation eld but may redirect it NOTE Scattering can be thought of as absorption of radiant energy followed by re emission back to the 39 quot field with quot quot 39 39 of energy Thus scattering can remove radiant energy of a light beam traveling in one direction but can be a source of radiant energy for the light beams traveling in other directions The fundamental law of extinction is the Beer Bouguer Lambert law which states that the extinction process is linear in the intensity of radiation and amount of matter provided that the physical state ie T P composition is held constant NOTE Some nonlinear processes do occur as will be discussed later in the course Consider a small volume AV of in nitesimal length ds and area AA containing optically active matter Thus the change of intensity along a path ds is proportional to the amount of matter in the path For extinction d1 3 1 ids 313 For emission 6 1 lids 314 where g 1 is the volume extinction coefficient LENGTH39I and J 1 is the source function In the most general case the source function J 1 has emission and scattering contributions Generally the volume extinction coefficient is a function of position s Sometimes it may be expressed mathematically as g 1 s but s is often dropped NOTE Volume extinction coefficient is often referred to as the extinction coef cient Extinction coefficient absorption coefficient scattering coefficient 5 51 33 315 NOTE Extinction coef cient as well as absorption and scattering coef cients can be expressed in different forms according to the de nition of the amount of matter eg number concentrations mass concentration etc of matter in the path 0 Volume and mass extinction coefficients are most often used Mass extinction coef cient volume extinction coef cientdensity UNITS the mass coef cient is in unit area per unit mass LENGTH2 MASS39I For instance cm2 g39l m2 kg39l etc If p is the density mass concentration of a given type of particles or molecules then g 2 05 p fa 316 lm 2 pk where the g ag and k A are the mass extinction scattering and absorption coefficients respectively NOTE L02 uses k A for both mass extinction and mass absorption coef cients The extinction cross section of a given particle or molecule is a parameter that measures the attenuation of electromagnetic radiation by this particle or molecule In the same fashion scattering and absorption cross sections can be de ned UNITS the cross section is in unit area LENGTHZ If N is the particle or molecule number concentration of a given type of particles or molecules then aw CLAN Li CLAN 317 lm 0a1N where 0399 1 013 and O39a 1 are the extinction scattering and absorbing cross sections respectively UNITS Particle number concentration is in the number of particles per unit volume LENGTH393 Optical depth of a medium between points s1 and s is de ned as 7452sri ezsds UNITS optical depth is unitless SI Sz NOTE same name optical depth optical thickness optical path 0 If g 1 s does not depend on position called a homogeneous optical path thus 616 lt eJgt and 7152 31 lt ax gt 32 31 lt ax gt S For this case the Extiction law can be expressed as I1 I0eXp6239 I0eXp6lt W1 gts 318 Optical depth can be expressed in several ways 71SISZ nds Ip 21ds Name s 319 o If in a given volume there are several types of optically active particles each with ied etc then the optical depth can be expressed as a 2 ma ds z wads 2 Hands 320 where p and N is the mass concentrations densities and particles concentrations of the ith species 4 Simple aspects of radiative transfer Let s consider a small volume AV of in nitesimal length ds and area AA containing optically active matter Using the Extinction law the change loss plus gain due to both the thermal emission and scattering of intensity along a path ds is dIA Tail1015 Ba139 Ads Dividing this equation by g 1 ds we nd 321 Eq 321 is the differential equation of radiative transfer called Schwarzchild s equation NOTE Both I 1 and J are generally functions of both position and direction The optical depth is 11s1s ji msds 1s1s Thus d7 5075 1s1s Using the above expression for dm we can rewrite Eq 321 as 2 11 J1 dig or as 322 amp 211 J1 dr These are other forms of the differential equation of radiative transfer Let s rearrange terms in the above equation and multiply both sides by exp 11 We have 6Xp 44 W 44 exp mm d21 and using that dIxexpxexpxdIxexpxIxdx we nd dI1 eXp 11 eXp r1J1d rl Then integrating over the path from 0 to SI we have 364115 eXp 1M1 s IeXp r1s1sJ1d r1 and 1451 1406XP TA510ireXP U51SJgdh 0 Thus 1451 1106Xp TSl0 TeXpTASISJ4de 0 and using 6 TA 2 evl S dS we have a solution of the equation of radiative transfer often referred to as the integral form of the radiative transfer equation 1131 110 XPTS10feXPTS1s113eads 323 NOTE i The above equation gives monochromatic intensity at a given point propagating in a given direction often called an elementary solution A completely general distribution of intensity in angle and wavelengths or frequencies can be obtained by repeating the elementary solution for all incident beams and for all wavelengths or frequencies ii Knowledge of the source function J 1 is required to solve the above equation In the general case the source function consists of thermal emission and scattering or emission from scattering depends on the position and direction and is very complex One may say that the radiative transfer equation is all about the source function gt Plane parallel atmosphere For many applications the atmosphere can be approximated by a plane parallel model to handle the vertical strati cation of the atmosphere Plane parallel atmosphere consists of a certain number of atmospheric layers each characterized by homogeneous properties e g T P optical properties of a given species etc and bordered by the bottom and top infinite plates called boundaries 0 Traditionally the vertical coordinate z is used to measure linear distances in the planeparallel atmosphere 2 scos6 where 0 denotes the angle between the upwaru uouuar auu ure uuecuou or propagation of a light beam or zenith angle and p is the azimuthal angle Using ds dzc0s the radiative transfer equation can be written as d14z6 gadz c059 14Z6 J4z6 Introducing the optical depth measured from the outer boundary downward as Mm J eJltzgtdz and using dTl 21ZdZ and y c0s0 we have d1 2 demw Mum 324 o Eq324 may be solved to give the upward or upwelling and downward or downwelling intensities for a nite atmosphere which is bounded on two sites T Upwardintensity 1 isfor 12120 orOSQS Z Downwardintensity I isfor lSlSOor7r2S6S7r using that c0s0l c0s7L3920 and c0s70 l 120 zzmp T02 10 12T w ax 114 Z I Ilmwxo j w 14T Z0 Bottom m 1 I t Am rm Planeparallel atmosphere NOTE For downward intensity p is replaced by p Lecture 4 Blackbody radiation Main Laws Brightness temperature Obiectives 1 Concepts of a blackbody thermodynamical equilibrium and local thermodynamical equilibrium 2 Main laws gt Blackbody emission Planck function gt StefanBoltzmann law gt Wien s displacement law gt Kirchhoff 5 law 3 Brightness temperature 4 Emission from the ocean and land surfaces Required reading S 25 1 Concepts of a blackbody and thermodynamical eguilibrium Blackbody is a body whose absorbs all radiation incident upon it Thermodynamical equilibrium describes the state of matter and radiation inside an isolated constanttemperature enclosure Blackbody radiation is the radiative field inside a caVity in thermodynamic equilibrium Blackbody cavity NOTE A blackbody cavity is an important element in the design of radiometers Cavities are used to provide a welldefined source for calibration of radiometers Another use of a cavity is to measure the radiation that ows into the cavity eg to measure the radiation of sun Properties 0tblackb0d13 radiation 0 Radiation emitted by a blackbody is isotropic homogeneous and unpolarized Blackbody radiation at a given wavelength depends only on the temperature Any two blackbodies at the same temperature emit precisely the same radiation A blackbody emits more radiation than any other type of an object at the same temperature NOTE The atmosphere is not strictly in the thermodynamic equilibrium because its temperature and pressure are functions of position Therefore it is usually subdivided into small subsystems each of which is effectively isothermal and isobaric referred to as Local Thermodynamical Equilibrium LTE gt A concept of LTE plays a fundamental role in atmospheric studies e g the main radiation laws discussed below which are strictly speaking valid in thermodynamical equilibrium can be applied to an atmospheric air parcel in LTE 2 Main radiation laws gt Blackbody Emission Planck function Planck function BT gives the intensity or radiance emitted by a blackbody having a given temperature o Plank function can be expressed in wavelength frequency or wavenumber domains as 2hc2 B T 25exphckBM 1 4391 2h173 BvT2N 42 c 6XphVkBT l 2h 3 2 Ra 43 eXph vckBT l where A is the wavelength 17 is the frequency v is the wavenumber h is the Plank s constant k1 is the Boltzmann s constant kB 138XlO3923 J K39l c is the velocity of light and T is the absolute temperature in K of a blackbody NOTE The relations between B T 3 T and BA T are derived using that VdVZIVdV Zl dd andthat lc17lv D BVTBAT and BVT2ZBT 39 behavior of Planck function 0 If 7 gt 00 or 17 gt 0 known as Rayleigh Jeans distributions 2k 0 BAT 4 T 44a 2k N2 BVT ff T 44b NOTE Rayleigh Jeans distributions has a direct application to passive microwave remote sensing For large wavelengths the emission is directly proportional to T o Ifk gt0or 17 gtoo 2hc 2 BAT 15 6XP 410 MkBT 45a 2hN3 N B C eXp hv kBT 45b Io w 1 1 sift Puma MEIMICE w mquot u a H I I m mpquot W VE L39BIGTH Figure 41 Planck function on loglog plot for several temperatures gt StefanBoltzmann law The Stefan Boltzmann law states that the radiative ux emitted by a blackbody per unit surface area of the blackbody varies as the fourth power of the temperature F n BT 039 T4 46 where 039 is the Stefan Boltzmann constant O39b 567le 0398 W m392 K394 F is energy ux W m39z and T is blackbody temperature in degrees Kelvin K and BT TBJTd gt Wien s displacement law The Wien s displacement law states that the wavelength at which the blackbody emission spectrum is most intense varies inversely with the blackbody s temperature The constant of proportionality is Wien s constant 2897 K um km 2897 T 47 where km is the wavelength in micrometers pm at which the peak emission intensity occurs and T is the temperature of the blackbody in degrees Kelvin K NOTE This law is simply derived from dBAdk 0 NOTE Easy to remember statement of the Wien s displacement law the hotter the object the shorter the wavelengths of the maximum intensity emitted gt Kirchhoff s law The Kirchhoft s law states that the emissivity 8quot of a medium is equal to the absorptivity AA of this medium under thermodynamic equilibrium 8 A 48 where 87quot is de ned as the ratio of the emitting intensity to the Planck function AxL is de ned as the ratio of the absorbed intensity to the Planck function For a blackbody 8x Ah 1 For a gray body ie no dependency on the wavelength 8 A lt 1 For a non blackbody 8quot AAlt 1 NOTE Kirchhoft s law applies to gases liquids and solids if they in TE or LTE NOTE In remote sensing applications one needs to distinguish between the emissivity 0f the surface e g various types of lands ice ocean etc and the emissivity of an atmospheric volume consisting of gases aerosols andor clouds 3 Brightness temperature Brightness temperature Tb is de ned as the temperature of a blackbody that emits the same intensity as measured at a given wavelength or frequency or wavenumber Brightness temperature is found by inverting the Planck function For instance from Eq4 l T C2 b 11n1 f1 11 49 where I is the measured intensity and C111911gtlt108Wm392sr391um4 Cz14388x104Kum I For a blackbody brightness temperature kinetic temperature Tb T 39 For natural materials T134 S T4 S is the broadband emissivity NOTE In the microwave region the Rayleigh iJeans distributions gives Tb S T However a is a complex function of several parameters see below 4 Emission from the ocean and land surfaces The ocean and land surfaces can modify the atmospheric radiation eld by a re ecting a portion of the incident radiation back into the atmosphere b transmitting some incident radiation c absorbing a portion of incident radiation Kirchhoff 5 law d emitting the thermal radiation Kirchhoff s law INCIDENT RADIATION REFLECTED RADIATION EMITTED RADIATION 3 ABSORBED RADIATION 3 TRANSMITTED RADIATION Conservation of energy requires that monochromatic radiation incident upon any surface L is either re ected lr absorbed la or transmitted It Thus IiIrIa It 410 1IrIiIaIiItIiRAT 411 where T is the transmission A is the absorption and R is the re ection of the surface In general T A and R are a function of the wavelength RAAA TA 1 412 Blackbody surfaces no re ection and surfaces in LTE from Kirchhoff s law AA 8L 413 Opaque surfaces no transmission RxAx1 414 Thus for the opaque surfaces 8M 1 RA 415 Emission from the ocean and land surfaces 0 In general emissivity depends on the direction of emission surface temperature wavelength and some physical properties of the surface In the thermal IR 4ymlt7tlt 100 um nearly all surfaces are efficient emitters with the emissivity gt 08 and their emissivity does not depend on the direction Therefore the intensity emitted from a unit surface area at a given wavelength is In 8 BATS In the shortwave region 01 pm ltlt 4 pm emissivity is negligibly small In microwave 01 cmlt7tlt 100 cm emissivity depends on the type and state of the surface NOTE Differences in the emissivity of ice vs water provide the basis for microwave remote sensing of seaice see Lab 2 Table 41 Emissivity of some surfaces in the IR region from 10 t012 um Surface Emissivit Water 0993 0998 Ice 098 Green grass 0975 0986 Sand 0949 0962 Snow 0969 0997 Granite 0898 Lecture 7 WWW regions Obiectives 1 Absorption coef cient and transition function 2 Gaseous absorption in IR 3 Gaseous absorption in the Visible and near infrared 4 Gaseous absorption in UV 5 Spectroscopic databases HITRAN Required reading L02 32 421 AdditionaVadvanced reading Kyle TG Atmospheric transmission emission and scattering Pergamon Press Oxford 1991 Rotman LS et al The HITRAN 2004 molecular spectroscopic database Journal of Quantitative Spectroscopy and Radiative Transfer V96 139204 2005 1 Absorption coefficient and transmission function Absorption coefficient is de ned by the position strength and shape of a spectral line where S in the line intensity and f is the line pro le SIkvdv and Ifv v0dv1 Units of the line pro le f LENGTH often cm Dependencies S depends on T fV V0 1 depends on the line halfwidth 0L p T which depends on pressure and temperature Path length or path is de ned as the amount of an absorber along the path Since the amount of the absorbing gas can be expressed in different ways a different measure of path length are used For instance if the amount of an absorber is given in terms of mass density mass path length sometimes called optical mass is u ifpsds Homogeneous absorption path when kv does not vary along the path gt optical depth is 1 kvu Inhomogeneous absorption path when kv varies along the path then 239 kadu NOTE In general EV depends on both the wavenumber and path length Table 71 Units used for the path length or amount of absorbing gases absorption coef cient and line intensity Absorbing gas Absorption coefficient Line intensity NOTE The product of the absorption coef cients and path length optical depth should be unitless Monochromatic transmittance Tv and absorbance AV of radiance along the path are defined as TV 2 6Xp TV AV l TV l eXp Tv 72 NOTE same name Transmission function Transmittance 2 Gaseous absorption in IR Main atmospheric gases absorbingemitting in IR C02 H20 03 CH4 N20 CFCs Each atmospheric gas has a speci c absorptionemission spectrum 7 its own signature Table 72 The most important Vibrational and rotational transitions for H20 CO2 03 CH4 N20 and CFCs Gas Center Transition Band interval v m Mum cm l H20 pure rotational 01000 15948 63 V2 P R 6402800 continuum far Wings of the strong 2001200 lines water vapor dimm 6139s HzO2 C02 667 15 V2 P R Q 540800 961 lo4 overtone and 8501250 10638 94 combination 2349 43 V3 P R 21002400 overtone and combination 03 1110 901 v1 P R 9501200 1043 959 V3 P R 600800 705 142 V2 P R 600800 CH4 13062 76 V4 9501650 N20 12856 79 v1 12001350 5888 170 V2 520660 22235 45 V3 21202270 CFCs 7001300 Continuum absorption by water vapor in the region from 8001200 cm391 remains unexplained It has been suggested that it results from the accumulated absorption of the distant Wings of lines in the far infrared This absorption is caused by collision broadening between H20 molecules called selfbroadening and between H20 and nonabsorbing molecules NZ called foreign broadening ABSORF TION 7n atmosphere Figure 71 Lowresolution infrared abSOIption spectra of the major atmosphen39c gases 3 Gaseous absorption in the Visible and near IR 0 Absorption of Visible and near IR radiation in the gaseous atmosphere is primarily due to H20 03 and C02 Table 73 Main Visible and nearIR absorption bands of atmospheric gases Gas Center Band interval v cm lz M1110 cm l H20 3703 27 25004500 5348 187 48006200 7246 138 64007600 9090 11 82009400 10638 094 1010011300 12195 082 1170012700 13888 072 1340014600 Visible 1500022600 C02 2526 43 20002400 3703 27 34003850 5000 20 47005200 6250 16 61006450 7143 14 68507000 03 2110 474 20002300 3030 33 30003100 Visible 1060022600 02 6329 158 63006350 7874 127 77008050 9433 106 93509400 13158 076 1285013200 14493 069 1430014600 15873 063 1475015900 N20 2222 45 21002300 2463 406 21002800 3484 287 33003500 CH4 3030 33 25003200 4420 220 40004600 6005 166 58506100 CO 2141 467 20002300 4273 234 41504350 N02 Visible 1440050000 Energy Curve for Blackbody in 6000 K Snlar Irradime Cum Oulside Almospherc Sum Irradiunct Curve a Sea Level E 15 E 3 10 LL n a UV u 12 m 14 2 x 12 Wuvelcnglh pm wow NOTE 4 Gaseous absomuon m UV Izcture 6 chall LhatEm lt Eult Evmlt 15 molecules dislodged only by photons having the large margins shortwave UV and X7 rays Electrons on the outermost orbits can be disturbed by the photons having the energies of UV and visible radiation gt these electrons are involved in absorptionemission in the UV and visible Both an atom and a molecule can have the electronic transitions Electronic transitions of a molecule are always accompanied by vibrational and rotational transitions and are governed by numerous selection rules To avoid very complicated calculations of electronic transitions numerous measurements of the absorption cross sections of the atmospheric atoms and molecules absorbing in the UV and visible have been performed in laboratory experiments 0 In general the absorption cross section varies with temperature NOTE Absorption crosssections can be determined in the laboratory using the Beer BouguerLambert law recall Lecture 2 In such an experiment from a measure of the light intensity in the absence of sample Io and in the presence of a sample I through a vessel of length 1 containing a known concentration N of absorbing gas one can obtain the absorption crosssections from eXpaNo 01 NOTE Recall Lecture 2 for a known absorption crosssection cm the absorption coefficient is calculated as ay1 0quotle where N in the number of molecules of a given gas per unit volume of air 0 Electronic transitions ie highenergy UV photons may cause various photochemical and photophysical processes Absorption of a highenergy photon AB h V gtAB may result in the following pl lIlldl and 39 pruce e Luminescence AB 9 AB hvl Ionization AB 9 ABJr e Quenching AB M 9 AB M represent any molecule that can carry away energy Dissociation AB 9 A B Chemical reaction AB C 9 A BC Absorption of UV radiation in the gaseous atmosphere is primarily due molecular oxygen 02 and ozone 03 Thermosphere Mesosphere Stratosphere Troposphere 16 Shuman unga 3 1017 CDI39INHIJLII 3 lonimtion x continuum H m a 6 a 10 3 band E u V 1049 g Huggins bands 3 ShumanRun e g 1040 bands 9 a Lyman O 7 2 2 10 Chappuis U bands E 103922 E Herzberg g 1 Dee continuum D lt 1039 1 03925 l l 1 l l l l a 1000 2000 3000 4000 3000 Wavelength A Figure 73 Spectral absorption crosssections of Oz and 03 NOTE a Bands of O2 and 03 at wavelengths lt 1 pm are electronic transitions b These absorption bands are relatively uncomplicated continua because practically all absorption results in dissociation of the molecule so the upper state is not quantized c Despite the small amount of 03 no solar radiation penetrates to the lower atmosphere at wavelengths lt 310 nm because of large absorption crosssections of 03 Table 73 Wavelengths of absorption in the solar spectrum UV visible by several atmospheric gases Gas Absorption wavelengths pm N2 lt 01 02 lt 0245 03 017 035 045 075 H2O lt 021 06 072 H202 hydrogen peroxide lt 035 N02 nitrogen oxide lt 06 N20 lt 024 N03 nitrate radical 041 067 HONO nitrous acid lt 04 HN03 nitric acid lt 033 CH3B1 methyl bromide lt 026 CFC13 CFC l l lt 023 HCHO formaldehyde 025 036 N02 absorb at 7tlt 06 pm but photodissociate at 7 lt 04 pm Lecture 5 Composition and structure of the atmosphere Absorptionemission by atmospheric gases Obiectives 1 Composition and structure of the atmosphere 2 Basic properties of atmospheric gases 3 Basic principles of molecular emissionabsorption 4 Spectral line shapes Lorentz Doppler and Voigt pro le 5 Absorption spectra of main atmospheric gases Required reading S 1315 321 3135 Additional reading CNES online tutorial Chapter 1 httpceoscnesfr8100cdrom00ceos1sciencedgdgconhtm Advanced reading McCartney EJ Absorption and emission by atmospheric gases John WileyampSons 1983 1 Composition and structure of the atmosphere 0 Atmospheric gases are highly selective in their ability to absorb radiation Each radiatively active atmospheric gas has a speci c absorption spectrum 7 its own signature Thus the abundance of gases in the atmosphere controls the overall spectral absorption Table 51 The gaseous composition of the atmosphere Gases by volume I Comments Constant ga ses Nitrogen N2 7808 Photochemical dissociation high in the ionosphere mixed at lower levels Oxygen 02 2095 Photochemical dissociation above 95 km mixed at lower levels Argon Ar 093 Mixed up to 110 km Neon Ne 00018 Helium He 00005 Mixed in most of the middle Krypton Kr 00001 1 atmosphere Xenon Xe 0000009 Variable gases Water vapor H2O 40 maximum Highly variable photodissociates in the tropics above 80 km dissociation 000001minimum at the South Pole Carbon dioxide CO2 00365 increasing 04 per ear Slightly variable mixed up to 100 km photodissociates above Methane CH4 000018 increases due to agriculture Mixed in troposphere dissociates in mesos Ehere Hydrogen H2 000006 Variable photochemical product decreases slightly with height in the middle atmosphere Nitrous oxide N2O 000003 Slightly variable at surface dissociates in stratosphere and mesos Ehere Carbon monoxide CO 0000009 Variable Ozone 03 0000001 00004 Highly variable photochemical origin Fluorocarbon 12 CF2C12 000000005 Mixed in troposphere dissociates in stratosphere 7O l 39 l 39 r I quot l 39 d 60quot j Mesosphere 50 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 7quot u 39 39 39 39 39 39 39 Stratosphere E 40 e x a v E J 3O std1976 I 39 trop R l subtropsum 39U btropwin l S 20 I subarcsum subarcwin lO 39 39 quot f 39 39 39 39 39 39 39 39 quot 39 39 39 Troposphere O l 1 1 1 L 1 J l a J I 1 i 190 200 210 220 230 240 250 260 270 280 290 300 Temperature K Figure 51 Temperature pro les of the standard atmospheric models often used in radiative transfer calculations Standard US 1976 atmosphere is representative of the global mean atmospheric conditions Tropical atmosphere is for latitudes lt 300 Subtropical atmosphere is for latitudes between 300 and 450 Subarctic atmosphere is for latitudes between 450 and 600 and Arctic atmosphere is for latitudes gt 600 Temperature lapse rate is the rate at which temperature decreases with increasing altitude F T2T1Z2Z1ATAZ 51 where T is temperature and the height 2 O Adiabatic process is of special significance in the atmosphere because many of the temperature changes that take place in the atmosphere can be approximated as adiabatic For a parcel of dry air under adiabatic conditions it can be shown that dTdz gcp 52 where cp is the heat capacity at constant pressure per unit mass of air and cp cV Rma ma 28966 gmole is the molecular weight of dry air and g 981 ms2 is the acceleration of gravity The quantities gcp is a constant for dry air equal to 976 C per km This constant is called dry adiabatic lapse rate gt The law of hydrostatic balance states that the pressure at any height in the atmosphere is equal to the total weight of the gas above that level The hydrostatic eguation dPZ dz pZ g 53 where pZ is the mass density of air at height 2 and g is the acceleration of gravity 0 Integrating the hydrostatic equation at constant temperature as a function of 2 gives P P0 eXP ZH 54 where H is the scale height H k3 T mg and m is the average mass of air molecule m 4 8096XlO3926 kgair molecule EL gases 2 Basic properties of altitude km l J l l 10 10 10 10 104 l mixing ratio Figure 52 Representative vertical pro les of mixing ratios of some gases in the atmosphere Some important properties of atmospheric gases gt Obey ideal gas laws Boyle s law V N lP at constant T and the number of gas moles u Charles s law V N T at constant P and u Avogadro s law V N the number of gas molecules at constant P and T The egua on of state says that the pressure exerted by a gas is proportional to its temperature and inversely proportional to its volume P V p R T 55 Where R is the universal gas constant If pressure P is in atmospheres atm volume V in liters L and temperature T in degrees Kelvin K thus R has value R 008206 L atm K l mor1 gt The amount of the gas may be expressed in several ways i Molecular number density molecular number concentration molecules per unit volume of air ii Density molecular mass concentration mass of gas molecules per unit volume of air iii Mixing ratios Volume mixing ratio is the number of gas molecules in a given volume to the total number of all gases in that volume when multiplied by 106 in ppmv parts per million by volume Mass mixing ratio is the mass of gas molecules in a given volume to the total mass of all gases in that volume when multiplied by 106 in ppmm parts per million by mass NOTE Commonly used mixing fraction one part per million 1 ppm 1x10396 one part per billion 1 ppb 1x10399 one part per trillion 1 ppt 1x103912 iv Mole fraction is the ratio of the number of moles of a given component in a mixture to the total number of moles in the mixture NOTE mole fraction is equivalent to the volume fraction NOTE The equation of state can be written in several forms using molar concentration of a gas c uv P c T R using number concentration of a gas N c NA P N T RNA or P N T k using mass concentration ofa gas q c mg P q T R mg Avogadro s number NA 602212xlO23 moleculesmole gt Structure of molecules Based on the geometric structure molecules can be divided into four types Linear molecules CO2 N2O C2H2 all diatomic molecules e g 02 N2 etc o Symmetric top molecules NH3 CH3CL Spherical symmetric top molecules CH4 Asymmetric top molecules H20 03 Rmzu39unzl and vimeiunzl mminns uf Lhe mulecules Molecules as a ngxd quanuzed rotator I l l o I I LINEAR DIATDMIC LINEAR TRIATDMIC 2 co ND N02 N20 l 0 K gt lt ASYMMETRIC TOP H20 03 I o Molecules as a quantizedvibrator DIATOMIC MOLECULES N2 02 co 4 4 TRIATOMIC MOLECULES symmetric bending antisymmetric N20 coz O MONO 97 Figure 54 Vibrational modes of diatomic and triatomic atmospheric molecules NOTE The number of independent vibrational modes called normal modes of a molecule with Ngt2 atoms are 3N6 for nonlinear molecules and 3N5 for a linear molecule 0 The geometrical structure of a molecule determines its dipole NOTE A dipole is represented by centers of positive and negative charges Q separated by a distance d the dipole moment Q 1 Example water molecule The unique way in which the hydrogen atoms are attached to the oxygen atom causes one side of the molecule to have a negative charge and the area in the opposite direction to have a positive charge The resulting polarity of charge causes molecules of water to be attracted to each other forming strong molecular bonds Table 52 Atmospheric molecule structure and dipole moment status Molecule Structure Permanent May acquire dipole moment dipole moment N2 N H N No No 02 O 0 O No No CO C 0 Yes Yes C02 0 m0 No Yes in two vibrational modes N20 N NO Yes Yes H20 A Yes Yes H 03 A Yes Yes 0 H O H O O H H No Yes V in two vibrational C modes H NOTE The structure of a molecule determines whether the molecule has a permanent dipole or may acquire the dipole The presence of the dipole is required for absorptionemission processes by the molecules 0 3 Basic principles of molecular emissionabsorption Review of main derlvin phvsicaly 39 ciples 0f 39 39 1 The origins of absorptionemission lie in exchanges of energy between gas molecules and electromagnetic eld 2 In general total energy of a molecule can be given as E Em Evib Eel Etr Em is the kinetic energy of rotation energy of the rotation of a molecule as a unit body about 1500 cm391 farinfrared to microwave region Evil is the kinetic energy of vibration energy of vibrating atom about their equilibrium positions about 500 to 104 cm391 near to farIR Eel is the electronic energy potential energy of electron arrangement about 104105 cm391 UV and visible Eu is translation energy exchange of kinetic energy between the molecules during collisions about 400 cm391 for T 300 K 0 From Ermlt E lt Eviblt Eel follows that i Rotational energy change will accompany a vibrational transition Therefore vibration rotation bands are often formed ii Kinetic collisions by changing the translation energy in uence rotational levels strongly vibrational levels slightly and electronic levels scarcely at all 0 Energy Em Evib and Eel are quantized and have only discrete values speci ed by one or more quantum numbers Not all transitions between quantized energy level are allowed they are subject to selection rules 3 Radiative transitions of purely rotational energy require that a molecule possess a permanent electrical or magnetic dipole moment 0 If charges are distributed symmetrically gt no permanent dipole moment gt no radiative activity in the farinfrared ie no transitions in rotational energy Example homonuclear diatomic molecules N2 02 02 has a weak permanent magnetic dipole and thus has a rotational transition in microwave 0 C0 N20 H20 and 03 exhibit pure rotational spectra because they all have the permanent dipoles C02 and CH4 don t have permanent dipole moment gt no pure rotational transitions But they can acquire the oscillating dipole moments in their Vibrational modes gt have Vibrationrotation bands 4 Radiative transitions of vibrational energy require a change in the dipole moment ie oscillating moment N N 2 no vibrational transition 02 symmetric stretching mode Hgt C0 single vibrational mode gt C 02 V1 symmetric stretching mode gt adiatively inactive 9 6 9 V2a two bending modes have same energy l l Vzb degenerated modes V3 asymmetric stretching mode gt radiatively active Figure 55 Vibrational modes of diatomic and triatomic atmospheric molecules see also Figure 54 NOTE Homonuclear diatomic molecules N2 and 02 don t have neither rotational nor Vibrational transitions because of their symmetrical structures gt no radiative activity in the infrared But these molecules become radiatively active in UV NOTE The number of independent vibrational modes of a molecule with Ngt2 atoms are 3N 6 for nonlinear molecules and 3N 5 for a linear molecule Both H20 and 03 have three normal band v1 v and V3 all are optically active CH4 has nine normal modes but only V3 and v4 are active in IR 5 Electronic transitions 0 Electrons on inner orbits close to the atomic nucleus can be disturbed or dislodged only by photons having the large energies shortwave UV and X rays 0 Electrons on the outermost orbits can be disturbed by the photons having the energies of UV and visible radiation gt these electrons are involved in absorptionemission in the UV and visible 0 Both an atom and a molecule can have the electronic transitions Electronic transitions of a molecule are always accompanied by vibrational and rotational transitions and are governed by numerous selection rules 4 Spectral line shapes LorentzI Doppler and Voigt pro le Atomic Absorption Emission Spectrum 0 Radiation emission absorption occurs only when an atom makes a transition from one state with energy Ek to a state with lower higher energy for emission Ek Ej hCV Absorption Emission Molecular AbsorptionEmission Spectra Molecular absorption spectrum is substantially more complicated than that of an atom because molecules have several forms of internal energy This is the subject of spectroscopy and quantum theory Three types of absorptionemission spectra i Sharp lines of nite Widths ii Aggregations series of lines called bands iii Spectral continuum extending over a broad range of wavelengths Fiw v Line WWW Band Continuous spectra wavelength gt Figure 56 Concept of a line band and continuous spectra Three main properties that de ne an absorption line central position of the line eg the central frequency 170 or wavenumber v0 strength of the line or intensity S and shape factor or profile f of the line 0 Each absorption line has a width referred to as natural broadening of a spectral line 0 In the atmosphere several processes may result in an additional broadening of a spectral line of the molecules 1 collisions between molecules referred to as the pressure broadening 2 due to the differences in the molecule thermal velocities referred to as the Doppler broadening and 3 the combination of the above processes Lorentz pro le of a spectral line is used to characterize the pressure broadening and is defined as a 7239 fLV V0 56 v v0 2 a 2 wherefv Va is the shape factor of a spectral line vo is the wavenumber of a central position of a line a is the halfwidth of a line at the half maximum in cm39l often referred as a line width 0 The half width of the Lorentz line shape is a function of pressure P and temperature T and can be expressed as aPTa0P 0 57 0 where 10 is the reference halfwidth for STP To 273K P1013 mb 10 is in the range from about 001 to 01 cm391 for most atmospheric radiatively active gases For most gases n12 NOTE The above dependence on pressure is very important because atmospheric pressure varies by an order of 3 from the surface to about 40 km o The Lorentz pro le is fundamental in the radiative transfer in the lower atmosphere where the pressure is high 0 The collisions between like molecules self broadening produces the large line widths than do collisions between unlike molecules foreign broadening Because radiatively active gases have low concentrations the foreign broadening often dominates in infrared radiative transfer Doppler pro le is defined in the absence of collision effects ie pressure broadening as 2 1 V V fDV V0 O D exp j 58 OLD is the Doppler line width V0 12 05D 2kBTm 59 c where c is the speed of light k1 is the Boltzmann s constant In is the mass of the molecule for air m 4 8XlO3923 g o The Doppler broadening is important at the altitudes from about 20 to 50 km Voigt pro le is the combination of the Lorentz and Doppler profiles to characterize broadening under the lowpressure conditions above about 40 km in the atmosphere ie it is required because the collisions pressure broadening and Doppler effect can not be treated as completely independent processes fVoigtVV0 IfLVV0fDVVdl a w 1 VV 2 510 ex dv 0D7Z3932 JV v0 2 052 p 05D foo NOTE The Voigt profile requires numerical calculations Nature of the Voigt profile 0 At high pressure the Doppler pro le is narrow compare to the Lorentz pro le so under these conditions the Voigt pro le is the same as Lorentz pro le 0 At low pressure the behavior is more complicated 7 a kind of hybrid line with a Doppler center but with Lorentz wings Absorption coef cient of a gas is de ned by the position strength and shape of a spectral line kwvSfv vo 511 where S in the line intensity and f is the line pro le Szjkavdv and IfV V0dvl Dependencies S depends on T fv v0 1 depends on the line halfwidth 0 p T which depends on pressure and temperature Optical depth due to gaseous absorption is de ned as a product of the absorption coef cient and the path length Because the amount of an absorbing gas may be expressed in a number of possible ways eg molecules per unit volume mass of molecules per unit volume etc different kinds of absorption coef cient may be introduced is such a way that the optical depth remains unitless Introducing a path length or amount of gas 14 we have TV J kayd 512 1 Most 39 used absorption coef cients km Volume absorption coef cient in LENGTH39I km Mass absorption coef cient in LENGTHZ MASS keg Absorption cross section in LENGTHZ Mass absorption coef cient volume absorption coef cientdensity Absorption cross section volume absorption coef cientnumber concentration Thus optical depth can be expressed in several ways Tv51sz 2 Thde 2 Tpkmmds TNkmavdS 513 51 51 Table 53 Units used for path length absorption coef cient and line intensity for Eqs5ll 513 Absorbing gas Absorption coefficient Line intensity Units of the line pro le f LENGTH often cm N20 CFng 0 Each atmospheric gas has a speci c absorptionemission spectrum 7 its own radiative signature HITRAN is a main spectroscopic data base that contains information eg intensity and halfwidth for a total of about 1080000 spectral lines for 36 different molecules httpwwwhitrancom gt Microwave region Molecule AbSOIption line Frequency GHZ HZO 22235 1833 02 about 60 11875 see Figure 34 gt Thermal IR region lt AasonmoN 7 Figure 57 Lowresolution IR absciption spectra of the major atmosphen39c gases Table 54 The most important vibrational and rotational transitions for H20 CO2 03 CH4 N20 and CFCs Gas Center Transition Band interval v m Mum cm39l H2O pure rotational 01000 15948 63 v2 P R 6402800 continuum far Wings of the strong 2001200 lines water vapor dimmers H2O2 C02 667 15 v2 P R Q 540800 961 o rtone and combination 8501250 10638 94 V3 P R overtone and 2349 43 combination 21002400 03 1110 901 v1 P R 9501200 1043 959 V3 P R 600800 705 142 V2 P R 600800 CH4 13062 76 V4 9501650 N20 12856 79 v1 12001350 5888 170 V2 520660 22235 45 V3 21202270 CFCS 7001300 NOTE Continuum absorption by water vapor in the region from 8001200 cm391 remains unexplained It has been suggested that it results from the accumulated absorption of the distant wings of lines in the far infrared This absorption is caused by collision broadening between H2O molecules called self broadening and between H20 and non absorbing molecules N2 called foreign broadening gt NearrIR and Visible reg39ons r H20 03 and co NOW 25 Energy CurVE mr Blackhody m 5000 K 2 u Solar lrrad nce Curve Oumide Almmphere T 50121 lrrudmncc Curve a Sea Level E f I s E 3 F I hl39Itun 0 04 o x 12 La 20 1 4 Wavelength um Table 55 atmosphenc gases Gas N1 lt0l 01 lt0145 02 017035 045075 1110 lt021 057072 H101 hvdxn gen uernxide lt 035 N01 nitrnzm nxide lt 05 lt014 N02 nitratemdical 041067 HONOnIXnIISaEid lt04 HNO 11 uric acid 3th med Vlbrmnide CFCla CFCll HO 139 maldehxde NOTI NO absorb a Alt o 6 pm butphotodlssocmte at Alt o 4 pm Lecture 17 Principles of multiple scattering in the atmosphere Radiative transfer eguation with scattering for solar radiation in a planeparallel atmosphere Obiectives 1 Concepts of the direct and diffuse scattered solar radiation 2 Source function and a radiative transfer equation for diffuse solar radiation 3 Single scattering or rst order scattering 4 Legendre polynomial expansion of the scattering phase function Required reading L02 34 61 Appendix E 1 Concepts of the direct and diffuse solar radiation 0 The solar radiation field is traditionally considered as a sum of two distinctly different components direct and diffuse I 101 Idif Direct solar radiation is a part of solar radiation filed that has survived the extinction passing a layer with optical depth 1 and it obeys the BeerBouguerLambert extinction law where Io is the solar intensity at a given wavelength at the top of the atmosphere and w is a cosine ofthe solar zenith angle 90 p0 c0s90 t Idir The direct solar ux is oFo XPT o 172 2 Source function and a radiative transfer e uation for the diffuse solar radiation Diffuse radiation arises from the light that undergoes one scattering event single scattering or many multiple scattering 90 quot35 EM w o k E Sin le sea in V g g 939 luv 3 Multiple scattering Recall Lecture 4 where we de ned the source function J G7 thermal j scattering B9 where jxa thermal is the thermal emission jimerma ay13101 and A scattering is the radiation source from multiple scattering Using the volume scattering coef cient 35 and the phase function Pu p u p we have s3 8 IMGWQG W 173 y 47239 NOTE Recall the scattering phase function P01 q u q ie the element of the scattering matrix P11 represents the angular distribution of scattered energy as a function of direction By the definition see Lecture13 it is normalized as 1 JPcos 9519 1 47 Q where D is the scattering angle c0s c0s939c0s0 sin939sin9 cosq39 p u u 1 u 2121 u212 cosq39 p Thus the source function for diffuse solar radiation may be written as two components erl a Jew IHutuw39P yuw39dy39d 39 4 0 1 17 4 F0Pa a0gt oexp7 0 where the no is the single scattering albedo and P is the scattering phase function NOTE In Eq174 the rst term on the righthand side shows that the phase function redirects the incoming intensity in the direction u p to the direction u p and the integrals account for all possible scattering events within the 411 solid angle 0 The source function for scattering Eq174 is more complicated than a thermal source function i It involves conditions throughout the atmosphere while the thermal source function depends on local conditions only ii The phase function Pu p u p may be a very complex function of the directions and in general state of polarization Recall the radiative transfer equation de ned in Lecture 3 for a planeparallel atmosphere d1 7 y dl a I rlt0 J4Tlt0 Thus using the source function for scattering we can write the radiative transfer equation for the diffuse radiation as omitting the subscript difin I 511035 in dr 1 Q 20 11 23135 Q dQ F0P O eXp L39 0 175 47 NOTE Eql75 is an integrodifferential equation To solve Eql75 one needs to know the scattering coefficient 35 absorption coefficient 33 and scattering phase function Pu q p q as a function of wavelength in each atmospheric layer Eql75 can be simpli ed if there is no dependency on the azimuth angle For azimuthally independent case we may define the phase function as I 1 2 I I I P IP d 176 27239 0 Using Eql76 we can write the azimuthally independent radiative transfer equation for the diffuse radiation 6g 1 i Salt17 l1ra 39P d F0P0expro 177 T 2 1 47239 gt To find a solution of the radiative transfer equation for diffuse radiation ie to solve Eql75 or 177 various approximate and exact techniques have been developed Approximate methods i Single scattering approximation this lecture ii Twostream approximations Lecture 18 iii Eddington and DeltaEddington approximations Lecture 18 Exact methods i Discreteordinate technique Lecture 20 ii Addingdoubling technique Lecture 21 iii MonteCarlo technique Lecture 22 A 3 Single scatterimr and the single scatterimr Iflight has been scattered only once the source function from Eq173 becomes a J 1 M p FoPUA 0 OCXPT 0 178 and using the solution derived in Lecture 3 of the radiation transfer in a planeparallel atmosphere bounded by two sides at 10 and E EI for upward intensity called re ected intensity 239 239 11r 117 exp 7 1H 2quot 239 7I6XP JIT dT and downward intensity called transmitted intensity 139 Inn Mo Ii0 exp 7 1 r r 1quot t r jexp gt Jm df I 0 I we can write the solution for diffuse radiation in the case of single scattering as T T T T 14r 14r exp 1 1 I 39 179a a r r r F0P o ojexp dr ILL 47239 r 0 t t T 14r 140 6Xp I 179b 1 a r r r 1quot F0P o ojexp dr 47f 0 0 Assuming that there is no diffuse downward radiation at the top of the atmosphere IW w 0 and no upward diffuse radiation at the surface ie no re ection from the surface ITTuqp0 1710 Then from Eql79ab for a nite atmosphere with optical depth 1 we nd the re ected and transmitted diffuse intensities a 0 I 0 F0 IT 0 W 47r0 P 0 01 exp r i L0 1711 I 0 and for p is NOT equaled to no Irww F Pwmwwwo exp T jiexp T 1712 47ru7u0 0 and for HFHO 1 a 139 F r 14 Hi 0 w Pk o worm lt00 exp 1713 4 7w 0 0 o In the case of single scattering the diffuse intensities are directly proportional to the scattering phase function NOTE Ifthe atmosphere is optically thin ie small optical depth 1 lt1 then amp F0 P 6 7 Eql7ll simpli es to IT0 47 u called the single scattering approximation and often used in remote sensing 4 I eoendre 39 39 mmsmsinn of the scatterimr phase function For many practical applications the phase function must be numerically expanded in Legendre polynomials with a finite number of terms N as N Pcos Z wal cos 9 1714 10 where D is the scattering angle c0s c0s939c0s0 sin939sin9 cosq39 p u u 1 u 2ml uzl2 cosq39 p and a is the expansion coefficients expressed as l 1 iIP00s Bcos dcos 10 1N 1715 71 NOTE Orthogonal propemes of the Legendre polynomials 1 I cos cos d 6059 Ofor 1 k 1 1 J mcosemwosewcoqe 2121for l k The rst few Legendre polynomials are gwen by PM 1 awn 1 xxx Sax 71 P x 55 7 3x P4x gen 7 30 3 P5x 63x5 7 70 15x Rewritin the radiative transfer equation in terms of 39 39 I egendre 39 39 39 Eq1714 can be expressed in the terms of associated Legendre polynomials N N Plt039lt039 Z Z wiszmP1 39cosmlt039lt0 1716 m o 1 m where l m 1m alm260mw lmN OSmSN and 50 is the Kronecker delta 50 1 for m0 and otherwise 50 0 In similar manner we may expand the diffuse intensity in the cosine series N 1T Zlm12005m 0 1717 m0 Using Eqs1716 and 1717 and the orthogonality of the associated Legendre polynomials the equation of the radiative transfer for the diffuse intensity Eq177 splits into N1 independent equations in the form d1 r m m N m m 1 m m y1 r 160m7 2 wlPl MP MI new lm 1 N 1718 a m m m 2wz B MB 0F0 eXpro lm m0gt 39 quot quot 39 39 case From Eq1716 the azimuthindependent phase function can be expressed as N P 2 WP MB 1719 0 For this case Eq1718 simplifies to omitting the superscript 0 for m0 d r a N 1 1 Mde JZ wlPl MP y Way 4 d1 2 10 71 1720 00 N 1 szPI MP 0F06Xpro 47239 10 EXAMPLE Ex ansitm 0 the Hen Greenstein scatterin hase unctitm in the Legendre polynomials The HenyeyGreenstein scattering phase function is a model phase function which is often used in radiative transfer calculations 1 g2 P cos HG 1g2 2g cos 32 g is the asymmetry parameter Let s take g05 representative of aerosols Legendre expansion N PHG cos 9 Z 3131 cos 9 and 10 211 2 jPHG cos B cos d cos If N0 gt 1 1 w PHGcos P0cos dcos PHGcos 1 dcos 2 1 1 71 l 2 NOTE w is always 1 because of normalization of the phase function 1 1 E j Pcos dcos 1 71 Thus in the case of one term in the expansion we have 0 3 PHGcos z Eqmcos 11 1 0 Plot below shows that using only one term gives poor approximation NOTE In all plots below the black curve shows the HenyeyGreenstein phase function and the red curve is for the Legendre expansion all as a function of cos If N1 gt 1 1 1 w IPHG cos P0 cos dcos PHG cos 1 d cos 2 1 1 71 l 2 2 1 1 w E I PHG cos P1 cos dcos PHG cos cos dcos 15 1 71 Thus in the case of two terms in the expansion we have 1 PHG cos m 2 waI cos 115cos still accuracy is not so good 10 We can continue by including more terms to get a desirable accuracy Lecture 27 Climate radiative forcing of gasesI aerosolsI and clouds Part 3 Obiectives 1 Radiative forcing impact of clouds Recommended reading L02 841845 86 Additional reading Stephens G L 2005 Cloud feedback in the climate system A critical review J Clim 18 237273 1 Radiative forcing impact of clouds Radiative forcing of clouds is defined is defined as a difference between the net uxes in clean and cloudy atmospheric conditions AF FMTOA I TOA pin lean where FCIMITOA is the net total SWLW flux at the top of the atmosphere or at the tropopause in the presence of clouds FclmTOA is the net total ux at the top of the clean atmosphere ie a reference atmospheric condition NOTE Cloud radiative forcing is de ned as the IPCC radiative forcing but it has different meaning Similar to Eq262264 cloud radiative forcing can be expressed as AFSWFLWF p12 where S WF is the shortwave solar component of radiative forcing LWF is the longwave thermal IR component of radiative forcing 5 2P UOA EMWHU p13 SW clean UW7Fdm MFdm M We NOTE seerugmsbelaw ctuud rathatuye fumng shuws huw mush shuns muthty thenetrathatuun and thus therrrute untheEarth energy balance pusuuye fumng useurs uyer the reguns where shuns aattu unerease the net energy untu the Earth system whtle the negatuye fvmng us uyer the reguns where shuns aattu deareasethe net energy The sutar and terrestnat pmpemes ufstuuds haye nffsemng effeats In terms hfthe energy balance hfthe planet In the tungwaye Lws1uudsgenera11yreduaethe rathatuun ermssmn tn syaae andthus result In a heatung uftheytanet whteun the sw shuns reduce the ahsurhed sutarrathatuun due tn a generatty hrgner athedu than the undertyung surface and thus result In a mulmg hfthe ytanet Thetatest resutts mm EREE unhaate that m the gtuhat mean shuns reduaethe rathatuye heatung uftheytanet Thts mulmg us a tunauun nfseastm and ranges fmm apprummately 13 tu zt Wm2 wht1ethese values may seem small they shuutdhe sumpared truth the 4 Wm2 heatung yrethated by a duuhhng hf atmusyhens mmemrauun ufcarbtm dnmde In terms ufhermsyhena ayerages the LW and SW stuud fumng tend tn hatanae eash utherun the unnter hermsyhere 1nthe summer hemrsyhere the negatuye sw stuud fumng dummates the yusutuye LW stuud furnng and the shuns resuttun a mulmg ISCCP Total stand Amnum 19334990 Figurele Tutat fraatuunat sluud suyer annual avemgedhm 19831990E7mp11ed usung data rm the Intematuunat Satelhte cluud chmatulugypruyeat ISCCP Numi m m m an m In so an a an so F39gul 212 LW claudfaxcmg mp andSW claudfaxcmg mm fax a myquot average fmm 1925725 mmquotde BREE Annual average net cloud radiative forcing 198586 determined from ERBE Net cloud forcing is the result of two opposing effects 1 greenhouse heating by clouds or positive forcing and 2 cooling by clouds or negative forcingc10uds re ect incoming solar radiation back to space The areas where cooling is the greatest are represented by colors that range from yellow to green to blue In some areas the effect of the clouds is to produce some warming as shown by colors that range from orange to red to pink Net Cloud Forcing from CERESTerra July 2000 quot0 T 20 80 data Lecture 21 WW Part 4 Principles of invariance Adding method Obiectives 1 Principles of invariance 2 Adding method Required reading L02 631 634 64 Advanced reading Van de Hulst Multiple light scattering Volume 1 chapter 4 Cambridge Univ Press 1980 1 Principles of invariance Recall the de nitions of re ection and transmission of a layer introduced in Lectures 18 19 If the solar ux is incident on a layer of optical depth 12 Rlt00lt00 nIT0ylto0Fo T o o I T W10Fo Or in the general case If0ltoj IRlt0lt0IW wwd dr 0 0 Ifr ltoj Tommi 0 1 51 W W 0 0 o The principle of invariance for the semi infinite atmosphere Ambartzumian 1940 the diffuse re ected intensity cannot be changed if a layer of nite optical depth having the same optical properties as those of the original layer is added see L02 632 gt The principles of invariance for a mite atmosphere Chandrasekhar 1950 1 The re ected upward intensity at any given optical depth 1 results from the re ection of a the attenuated solar uX uOFO eXp r 0 and b the downward diffuse intensity at the level 1 F 1 t t t 39 1lr exp r0Rr1 ro2jRr1 rm r mud 211 0 0 p w u Rel 2M Rel mm 11 2 The diffusely transmitted downward intensity at the level 1 results from a the transmission of incident solar uX and b the re ection of the upward diffuse intensity above the level 1 F 1 1lrr u TTo2IRTITTd 212 0 Po 0 TTgtgto Rr39 E P P 11 3 The re ected upward intensity at the top of the nite atmosphere 1 0 is equivalent to a the re ection of solar flux plus b the direct and diffuse transmission of the upward diffuse intensity above the level 1 F 1 HowRryyo2jfm wmdy1Trexp ry 213 0 Omu H u RTo T 1 39 4 The diffusely transmitted downward intensity at the bottom of the nite atmosphere 111 is equivalent to a the transmission of the attenuated solar ux at the level 1 plus b the direct and diffuse transmission of the downward diffuse intensity at the level 1 from above F 1 Mary eXP T0T71 To 21ml er1lrrd 0 214 I nimexpwl 7 my H TTl T Idau T71 Tuuuu 1 u u 2 Adding method Adding method is an exact technique for solving the radiative transfer equation with multiple scattering It uses geometrical raytracing approach and the re ection and transmission of each individual atmospheric layer Strategy knowing the re ection and transmission of two individual layers the re ection and transmission of the combined layer may be obtained by calculating the successive re ections and transmissions between these two layers NOTE If optical depths of these two layers are equaled this method is referred to as the doublingadding method Consider two layers with re ection R1 and R2 and total direct plus diffuse transmission f1 and f2 functions respectively Let s denote the combined re ection and total transmission functions by R12 and in and combined re ection and total transmission functions between layers 1 and 2 by U and respectively OF J fR w 2sz f f1 T1R2R1 T1R2R1R2R1 TNTN 1 The combined re ection function R12 is R12 R1 flefl fleRlefl fleRleRlefl R1 flefl1 RI2 R1R22 215 R1 szlz 7 Rleyl NOTE In Eq215 we use that 1 2xquot x quot0 The combined total transmission function in is T f1 fleleZ fleRlelez flfzu 121122 R1R22 216 f1f21 R1R271 The combined re ection function U between layers 1 and 2 U RZ TNlRZRlRZ fleRleRle flR21 R112 1111222 217 flea 111122 1 The combined total transmission function Bbetween layers 1 and 2 5 fl fleR1 fleRleRl fl1 R112 R1R22 218 flu 111122 1 From Eqs215218 we find that R12 R1 flU fu T25 U R25 219 Let s introduce S R1R21 R12 71 Using that T T eXp ru39 from Eqs 218219 we find DDeXp rlun 1STl eXp L391LID1ST1 SeXp 11 0eXp11 10 2110 Practical application of least square theory Function Fitting Inverse problem I want to compute SST anomalies Pacific SS 250 200 150 100 Which mean do I remove OPTION 1 Remove the constant spatial mean Mean 2 135 C Function Fitting Pacif I SST Function Fitting How to remove the gradient with least square 250 200 150 100 250 200 150 100 1 Least Square Solution What if we want to remove the gradient associated with the California Current 250 200 150 100 250 200 150 100 O 5 10 15 20 25 T 2 Weighted Least J y EXgt Wy EX Square Solution 2 ETWE 1ETWy Remove the meridional gradient with different wei htin g 60 I we V 7 1 50 40 250 200 150 Equally weighted California Current and Gina Sea 250 200 150 100 5 10 15 20 25 250 200 150 100 Anomalies with different weighting 250 200 150 100 What if we want to remove the gradient associated with the South Cina Sea What if we want to remove the gradient associated with the South Cina Sea Lecture 16 Light scattering and absorption by atmospheric particulates Part 4 Scattering and absorption by nonspherical particles Obiectives 1 Types of nonspherical particles in the atmosphere 2 Raytracing method 3 TMatriX method 4 FDTD method 5 DDA method Required Reading L02 53 55 AdvancedAdditional Reading L02 54 Mishchenko Hovenier and Travis Eds Light scattering by nonspherical particles Academic Press 2000 Good website with information on various methods and numerical codes for scattering by nonspherical particles httpwwwt matrixde 1 Types of nonspherical particles in the atmosphere Aerosols Dry salts e g dry sulfates nitrates seasalt cube or rectangularlike shapes Dust various shapes Carbonaceous chainlike aggregates Bio aerosols various shape Ice crystals several typical habits Excess vapor pressure over ice hPa 22 Figure 162108 crystal habits I l 4 6 1 2 1 0 Temperature 0 mmmrrmhsad o In contrast to the spherical particles the scattering properties of nonspherical particle also depend on shape of the particle and its orientation with respect to the incident light beam F or nonspherical particles Recall Lectures 1415 In the far eld zone ie at the large distances r from a particle the solution of the vector wave equation can be obtained as E15 eXp ikr ikz52 S3 E LEfJ W 4 51 LEiJ and for the Stokes parameters 1 10 Q oquot P Q U 47272 U0 V V where P is the phase matrix N m 4 9 go gm u m go go u m go go u m go go u 161 N m 4 Orientation of the Qarticles Aerosol particles have random orientation in space whereas the ice crystals are often oriented Orientation averaged scattering phase function and scattering cross section are 1 2 Ir2 I P J I P39a39y39ax39a39y39smaklah39y39 162 27239UJ 0 0 1 2 Ir2 I a U39 a39 39 sma39da39d 39 163 s 2 i 7 7 where oc and if are the orientation angles of a nonspherical particle with respect to incident light beam 2 Raytracing method Raytracing method or geometrical optics approximation or ray optics approximation is an approximate method for computing light scattering by particles much larger than a wavelength ie the smallest size parameter is about 80100 Basic principles Raytracing method is based on the assumption that the incident EM wave can be represented as a collection of independent parallel rays Ray tracing is commonly performed using a Monte Carlo approach Ray tracing consists of two parts 1 diffraction theory for the forward scattering peak 2 ray tracing using Fresnel re ection and transmission formulas Advantgges Raytracing method can be applied to any shape spherical or non spherical Limitations Raytracing method is an approximate by de nition Limited range of size parameters Absorbing particles require special treatment gt Di raction In geometric optics light may be treated as rays except for Fraunhofer diffraction around a particle Px y 2 gt OBSTACLE a r A w F xyz g 57 A r Babinet s principle diffraction pattern is the same from an aperture as for opaque APERTURE particle of same size Integrate the far eld contribution of incident wave over the aperture iE0 Rli exp iszdA 164 E7 Huygens principle 7 each point is a source of circular wave fronts For sphere circular aperture the diffraction pattern is 19 712sz 4 165 10 2J1xsin 92 xsin where k 271 and J1 is Bessel function Examgle scattering diagram for diffraction by a circular disk Diffraction peak in phase function width 9 NlX height PO X2 First zero at X sin 383 and max at X sin 514 Lu x I r 08 0 2 C LU 39 5 06 I7 3 3 3 04 4 2 Z O 2 02 n o 2 3 4 5 e 7 e 9 IO gt Fresnel re ection and tranmlim39M Q 46quot Halo 2 Sundngl at mienquot my Scattered my Snell39s law gives refraction angle 6 sinei msin9t Fresnel formula for polarized re ection amplitude coef cients cos 0 lmz isinz 0 ry cos 0 1lmz 7 sinZ 0 lm2 7 sin 2 91 r m2 cos 91 2 39 2 2 lm 7 sin 91 m cos 91 r Re ectivity and transmission coeffi cient for intensity Z rl Tr1 R Tl1 Rl r 166 167 168 169 2 NOTE For 600 re ection is R H gt re ectivity increases with refractive m index gt 017M611 w m39mt Mwmn39mt transmission 8vluncyion peak i 39H a t rac m u 2 L m a f quot3quotan a refvuclv39on Scanering Angle 8 w39 I 1 1an randomly onmted hexagonal me crystals from Mon 1992 3 TeMalrix method TVMaLrix meLhodTM1VL enables calculahon ofopueal propemes ofpamcles thh rotauonally symmeme shape such as elhpsoxds mular cyhhders Chebyshev shapes etc NOTE FORTRAN code of TM is avaxlable at hnp www glss hasa govNanmm Basic Qrincigles TMM is based on expanding the incident EM and scattered elds in vector spherical wave functions The T matrix transforms the expansion coef cients of the incident eld into those of scattered eld and if known can be used to compute any scattering characteristic of a nonspherical particle The elements of the T matrix are independent of the incident and scattering elds and depend only on the shape size parameter and refractive index of the scattering particle and on its orientation with respect to the reference frame Advantages TTM is highly accurate and computationally fast Limitations Limited types of particle shapes Limited range of size parameters xlt 30 Agglications Mishchenko et al Tmatrix is applied to a mixture of ellipsoids dustlike ice crystal like Mackowski et al modi ed Tmatrix is applied to fractallike sphere clusters soot aerosols 4 Outline of the FDTD method Finite Difference Time Domain FDTD method enables calculations of optical properties of particles of complicated geometries and compositions Basic Qrincigles FDTD solves the Maxwell s curl equations in the timedomain by introducing a nite difference analog The space containing a scattering particle is dicretized by using a grid mesh The existence of the particle is represented by assigning suitable electromagnetic constants in terms of permittivity permeability and conductivity depending on particle properties over the grid points anificla Advantages FDTD can be applied to particles having any shape and composition Limitations Known implementation problems for instance staircasing effect due to selection of Cartesian mesh grid Limited range of size parameters up to X 1520 Agglicatians ice crystals Yang et al FDTD for size parameter N15 and raytracing for x gt15 NOTE see Chapter 54 in L02 5 DDA method Basics of dipole interactions In the Rayleigh limit perpendicular and parallel components of scattered intensities in the far eld are I IU39kADtZ rZ 1 Int 4th cos Z rZ Where 1 is the polarizability of the particle and it relates to the refractive index via LorentzLorentz formula also known as ClausiusMossotti formula see Lecture 13 EM dd 0 twu ixnlated Figure 164 Two isolated dipoles scatter the incident EM eld into all directions p An observer at point 1 will measure the superposition ofscattered waves oftwo dipoles propagating into the direction ofobservation ie in scattering angle 6 The interference h 1 1 u L 1quot AL I difference The phase quot 39 length oftwo waves A5 xlr cos 6 where x is the size pararneter Thus the scattered eld is Em N E1 explt fir E2 explt 45 A5 and the detected intensity averaged over the full cycle IH2 s E3 E 2EE2 cos M 115552 A57r 37 57 gt elds cancel out ofphase A5 4 27 41g gt elds reinforce in phase NOTE The forward scattering waves are alvmys in phase 1610 1611 1612 In general case for nonspherical particle represented by many dipoles the phase difference depends on both the distance between dipoles and the direction of scattering except for 9 00 ie forward scattering gt the scattered radiation is a complex superposition of individual scattered waves of many different phase differences The larger the particle the higher intensity scattered in the forward direction and the greater the forward to backward asymmetry 1 I I I une dipole 0 m T q two dipoes four dipnles h 06 m c Q E 04 39 quot quot I quot39 39 0 l l L 0 30 60 90 I20 I50 I30 Sca ering Angle degrees Figure 165 Intensity of dipoles lied on one line at the distance of one wavelength and interacted with each other Bohren 1987 1 o In addition to phase differences the scattered eld is affected by interaction of dipoles with each other Consider a particle composed of many dipoles The scattered eld is incident eld plus the elds produced by each dipole 15 Z 1613 The dipole moment of j th dipole is p aJE 1614 dipole 1 where xj is the polarizability of the dipole and E is he eld acting on the dipole mate 1 which is the superposition of incident eld and the elds caused by other dipoles Thus P aEW1ZA pk 1615 1 where 2 A1 p k is the contribution from the electric eld at jth dipole from the kth 1 dipole NOTE Solving Eqsl6l3 l6 15 for all pj lies the basis ofthe Discrete Dipole Approximation DDA method Outline of the DDA method DDA Discrete Dipole Approximation method enables computation of optical properties of arbitrary shaped inhomogeneous and anisotropic particles NOTE DDSCAT is a FORTRAN implementation of the DDA technique The code and user guide are openly available at httpwwwastroprincetonedudraineDDSCAThtml Basic principles In DDA the particle is replaced by an array of polarizable points dipoles and then the electromagnetic scattering problem for an incident periodic wave interacting with this array of point dipoles is solved exactly Incident Field o39o39o39 o o O O 9 hf o o onE Mun 5 E as 9255 1 9r 0 i 1 o g 1239 we 1 i i i Q C n u39t 3911 lg a lSKU X3BBB Hm I Figure 166 A spherical particle is represented by individual dipoles in DDA calculations Advantages DDA can be applied to particles having any shape and composition ie homogeneous or aggregates Agglicabilit and limitations 0 DDA DDA is completely exible regarding the geometry of a particle being only limited by the need to use an interdipole distance d small to satisfy 2 m d lt 1 1616 Where m is the complex refractive index of the particle If a particle of volume V is represented by an array of N dipoles located on a cubic lattice with lattice spacing d then V N613 1617 The size of the particle can be characterized by the effective radius amas def 3V47r 1618 ie amis the radius of an equal volume sphere 2a Then the size parameter is x 7614f and it can be related to N as N 2 1619 m d le 76204 4 J 10 lml The number of dipoles N 11173 so as particle size increases the ver large number of dipoles are required Therefore DDA is limited by the size parameter of about 1520 or the number of dipole is up to N106 1 17 g 01 r E G E 011 O01 7 1768500265i sz 5 17904 a or g 1 80A 7 K o a 560 H W am 7554 l l l l l l l l l l l l H 3 1064 error in Qh39 m l l 2320 w llllillllillllillllillllillllillllillllilllli lllillllillllillll 0 1 2 3 4 5 6 7 B 9 10 11 12 13 x o l Figure 167 Scattering and absorption efficiencies for a sphere with m133001i The upper panel shows exact calculations from Mie whereas the middle and lower panels show fractional errors in Q5 and Q2 calculated with DDSCAT for different numbers of dipoles N Draine and Flatau 1994 Applications Kalashnikova and Sokolik calculations of optical properties of mineral dust particles Random shape1 Random shape 2 Random shape 3 10000 i 39 39 Sphere Ellipse Random shar J cdgo Rectangular disk HmrycyGrecnslcin 1000 100 0 10 393939uW3939 001 i i l i l 0 50 100 150 Scattering angle 0 degree Figure 168 Upper panel Representation of different shapes of dust particles for DDA calculations Lower panel Scattering phase function calculated with DDA for a log norrnal size distribution with re 05 pm at 9 05 pm n1 151 i0002 From Kalashnikova and Sokolik 2002 Lecture 11 WW IR radiative heatingcooling rates Obiectives 1 IR radiative transfer revisited 2 Infrared radiative heating cooling rates in the cloudfree atmosphere 3 Concept of the broadband ux emissivity Required reading L02 422 45 47 1 IR radiative transfer revisited Recall Lecture 8 where we have derived the solutions of the radiative transfer equation for the monochromatic upward and downward intensities in the IR for a planeparallel atmosphere consisting of absorbing gases no scattering L39 L39 1Jr Bvr exp T 1 I 83a J6Xp Bvr39dr39 13r ijexm T 1Br dr 83b 0 and in terms of transmission function 1Jrm BrTr r x 84a I BAMJTVU lama r d1quot r L w r 1jr j Bvr VT T udr 84b 0 d1quot Recall that Eq83a b and Eq 84a b have been derived for the entire atmosphere with the optical depth TV for two boundary conditions Surface assumed to be a blackbody in the IR emitting with the surface temperature Ts T IV Top of the atmosphere TOA 1 0 no downward emission 130 0 In Lecture 3 the upwelling and downwelling uxes were de ned as 1 FVT 27239JIJuudu 111 0 1 t t FV 2 I1Vd 0 NOTE Eql 11 assumes that there is no dependency on q in a planeparallel atmosphere Thus we can rewrite the radiative transfer equation and its solutions in terms of the monochromatic upward and downward uxes From Eq83a b we have 1 T T FRI MBA 1 exp Td 0 1 1 TT 112a 272 jexp BVI dI du 1 and 1 r T Tr I I FV T27FJ SKEWT BVTde 112b 0 0 Let s introduce the transmission function for the radiative ux called diffuse transmission function or slab transmission function or ux transmission function as 1 Mr 2 j TVTd 1131 0 where T V cp is the monochromatic transmittance de ned in Lecture 7 Eq 72 Spectral diffuse transmission function or transmittance may be de ned as l vaT 2 TvTd 114 0 Using the de nition of monochromatic diffuse transmittance and solution of the radiative transfer equation expressed via the transmittance Eq 84a b the solution for uxes can be written as Ffm zBVoUTHH r ll5a I 1 f I 3VT39W Td7 1 d7 and 1 T Mfg 1 F T 7TB T V d l39 v 1 V d1 115b NOTE On the right side of Eql 15a for the upward ux the first term gives the surface emission that is attenuated to the level I and the second term gives the emission from the atmospheric layers characterized by the Planck function multiplied by the weighting function dva d7 Likewise the downward ux at a given layer Eq ll5b is produced by the emission from the atmospheric layers I I A I 2 Infrared radiative 39 rates in the cloud free Radiative processes may affect the dynamics and thermodynamics of an atmosphere through the generation of radiative heatingcooling rates The radiative heating or cooling rate is de ned as the rate of temperature change of the layer dz due to radiative energy gain or loss 1 anet dt cpp dz dF net dp 116 i 0P where CI is the specific heat at the constant pressure c1 100467 JkgK and p is the air density in a given layer NOTE The thermodynamic equation for the temperature changes in the atmosphere ie the rst law of thermodynamic for moist air includes the radiative energy exchange term ie total radiative heatingcooling rates which are a sum of solar and infrared heating cooling rates In this lecture we discuss IR radiative rates only solar will be discussed later in the course Recall the de nition of the monochromatic net ux net power per area at a given height see Lecture 3 Fvltzgt F3 2 F 2 117 Similar we can de ne the total net ux FzFTz F z 118 Introducing the net ux F zAz at the level zAz see gure below we nd the net ux divergence for the layer Az as AF FzAz Fz F zAz lt F z hence AF lt 0 gt a layer gains radiative energy gt heating F zAz gt F z hence AF gt 0 gt a layer losses radiative energy gt cooling FT 2 A2 Flz A2 T pAp l zAz uAu T P L z u Flz Flz 0 0 p5 WWWWWW EXAMPLE Calculate longwave cooling at night for an atmospheric layer from 0 to 1 km using the upwelling and downwelling uxes calculated with MODTRAN for US Standard Atmosphere 1976 Altitude IR Upwelling ux IR Downwelling ux km Wm Wm 0 390 285 1 375 250 SOLUTION Need to nd net uxes at each altitude Fz Flz Flz At 0 km Fuel 390 7 285 105 Wm2 At 1 km Fuel 375 7 250 125 Wm2 Thus AF 20 Wm2 LIT l dFmt 20Js 1m 2 R Cpp dz 117kg m31004 Jkg lK 11000 m 7 dTdt 17x10 5 Ks 15 Kday Thermal infrared reg39on Each part of the atmosphere emits and absorbs radiation Where absorption dominates there is the net radiative heating where emission dominates there is net cooling The latter is more common Thus one can talk about IR radiative cooling rates giving positive values of or IR radiative heating rates giving negative values of dz IR LT z To calculate the IR heatingcooling rates one needs to know i Pro les of IR upwelling and downwelling uxes to calculate the pro le of the IR net uxes To compute the IR downward and upward uxes one needs to know 1 Atmospheric characteristics vertical pro les of T P and air density and 2 The vertical pro les of IR radiatively active gases clouds and aerosols ii Using the pro le of net uxes and air density one calculates the IR radiative heatingcooling rates as l dFz dt IR cpp dz E tect of the varving amount of a gas on IR radiation under the same 39 condition Consider the standard tropical atmosphere and dry tropical atmosphere same atmospheric characteristics except the amount of H20 see Figure below 10 9 8 E 7 x 6 g 5 g 4 E 3 2 1 0 0 5 10 15 20 H20 glkg Calculated IR uxes for tropical dotted lines and dry tropical atmospheres solid lines 10 9 8 E E7 I Fup w D Fdw 395 5 3 4 Fup dry 3 3 Fdw dry 2 1 O 60 160 260 360 460 Flux Wm2 lt H20 increases in a layer gt F i increases because more IR radiation emitted in a layer gt F isurface increases H20 increases in a layer gt F T decreases because more IR radiation absorbed but reemitted at the lower temperature gt F TTOA decreases Increase of an IR absorbing gas contributes to the greenhouse effect Calculata l Iant was rm nupical dnma l lines and n dryuwical anquotng 1mm lines Amman In mm mm Thelarga39 changes ume net ux mm mama m annLhErx e Lhelarger slaps umz V Z melarga IR cuuhng muss Calculata l Ichnlhg mes fur nupical dnma l lines and an mind anquotng snlid lines Altitudekm Ogmwbwmuoocoo A K 71 0 Hmu39xg rates Krlzy NOTE The largest 1R cuulmg mes fur me standard mm atmnwhere are unturned m me surface layer IR cw l39ng may ocx39ndivilual was B Amman km S r i 72 1 Lungwavn Ramauvo Healmg 1 6 per Day Figure 111quotnn mermal m a39m thr represent cooling NOTE v A near 15 km her a emu am um of h anquot levels 3 and 10 Ian the stringer ofthese being associated with the higher altitude peak falls offrapidly with height Consequently the atmosphere in the upper troposphere and band nire o e strong absorption lnn with the peak warniing between 20 and 30 km This heating is due to the absorption urn band IR ranting ratex in m rent thulb e WM Altitude ka I a 05 73 725 72 4 s 4 Longwave Radiative Heating c per Day Figure 112 The total 13 heating rates pro les calculate for three different model atmospheres 3 Concept of the 39 quot 39 ux The broadband ux emissivity approach allows calculation of infrared uxes and heatingcooling rates utilizing the temperature in terms of the StefanBoltzmann law instead of the Planck function Based on Eql 15 a b the total upward and downward uxes in the path length u coordinates may be expressed as Flat i 7IBVTJTVfudV co 1 f ll9a j I BVTu dTVu udu dv 0 0 d I and 00 u f r l r dTv u u r F 11 I nBVTuTdudv 1191 0 1 1 From the StefanBoltzmann law see Lecture 3 we have J BVTdv UBT4 0 Let s define the isothermal broadband emissivity as man Tu dv sfuT0 1110 4 GET Using the isothermal broadband emissivity Eql 19a b may be approximated as Flat O39BT54l gfuTx O T4ud8fu u39Tu39du lllla B du 0 Lecture 4 MW Sun as an energy source Solar spectrum and solar constant Obiectives 1 Concepts of a blackbody thermodynamical equilibrium and local thermodynamical equilibrium 2 Main radiation laws gt Planck function gt StefanBoltzmann law gt Wien s displacement law gt Kirchhoff s law 3 Sun as an energy source Required reading L02 12 143 Appendix A 2 1 Concepts of a blackbody and thermodynamical equilibrium Thermodynamical equilibrium describes the state of matter and radiation inside an isolated constanttemperature enclosure Blackbody is a perfect absorber emitter of radiation Properties 0tblackb0d13 radiation 0 Radiation emitted by a blackbody is isotropic homogeneous and unpolarized Blackbody radiation at a given wavelength depends only on the temperature Any two blackbodies at the same temperature emit precisely the same radiation A blackbody emits more radiation than any other type of an object at the same temperature NOTE The atmosphere is not strictly in the thermodynamic equilibrium because its temperature and pressure are functions of position Therefore it is usually subdivided into small subsystems each of which is effectively isothermal and isobaric referred to as Local Thermodynamical Equilibrium LTE 2 Main radiation laws Planck function BT gives the intensity or radiance emitted by a blackbody having temperature T NOTE Distribution of blackbody radiation as a function of wavelength known as the Planck law cannot be predicted using classical physics This derivation requires quantum mechanics See L02 Appendix A for the derivation of the Plank function Plank function can be expressed in wavelength frequency or wavenumber domains as 2hc 2 B T 25exphckBT1 1 4391 2h173 B T V 02eXph17kBT l 4392 3 2 RUFL 43 eXphvckBT l where A is the wavelength 17 is the frequency v is the wavenumber h is the Plank s constant k1 is the Boltzmann s constant kB 13806XlO3923 J K39l c is the speed of light and T is the absolute temperature of a blackbody NOTE The relations between B T 3 T and BA T are derived using that BTdl7 BVTdV 31056 andthat l c17 lv gt BAT i ZBT and BVT 12310 5 mm mum w quotswam 5 a m x mmmm u m rigm u Planck imam m lngrlng W m several temperatures Dawn Hand umL in am n V 7gt 0 knan asRaylzigh 4m ammns 2m A 2m2 72 3 T 5 NOTE Thslmgwave 1mm has a ma appucaumm passm micmwavz mum mug lfrgt0nr i am 2 817 iexpww MM 1 zw z 8V Sith 7 UCBT 4 6 4 7 Toward the ult ramlit calaslie ph RayleighJeans Law Equot a 5 E E 1 g Planck Law E I I Cum s agree at Every low frequencies Freq uenSy Figure 42 Plank function and its asymptotic behavior Stefan Boltzmann law The total power energy per unit time emitted by a blackbody per unit surface area of the blackbody varies as the fourth power of the temperature F 1 BT 0 T4 48 where 039 is the Stefan Boltzmann constant 0quot 5671X108 W m392 K394 F is the radiant ux W m39z and T is blackbody temperature K and BTB Td Wien s displacement law The wavelength at which the blackbody emission spectrum is most intense varies inversely with the blackbody s temperature The constant of proportionality is Wien s m 2897 T 49 where 7cm is the wavelength in micrometers pm at which the peak emission intensity constant 2897 K um occurs and T is the temperature of the blackbody in degrees Kelvin K NOTE This law is derived from dBAdh 0 NOTE Easy to remember statement of the Wien s displacement law the hotter the object the sh 0rter the wavelengths of the maximum intensity emitted Kirchhoft s law The emissivity 51 of a medium is equal to the absorptivity A7 of this medium under thermodynamic equilibrium ax 7 410 where 87quot is de ned as the ratio of the emitted intensity to the Planck function AxL is de ned as the ratio of the absorbed intensity to the Planck function For a blackbody 8h Ax 1 For a non blackbody 8quot A lt 1 For a gray body ie no dependency on the wavelength 8 A lt 1 NOTE The Kirchhoft s law applies to gases liquids and solids if they in TE or LTE o For atmospheric radiation transfer applications one needs to distinguish between the emissivity 0f the surface e g various types of lands ice etc and the emissivity of an atmospheric volume consisting of gases aerosols and or clouds gt Fmiqqivitv 0f the I L volume Absorption and thermal emission of the atmosphere volume is isotropic Kirchhoff s law applied to volume thermal emission gives jltherma1 alB1T 411l where 33 is the absorption coefficient of the atmospheric volume and j A is the thermal emission coef cient which relates to the source function J 3 introduced in Lecture 3 as J A jA thermal39l39 jA scattering Be and 3939 is the extinction coefficient of the atmospheric volume Recall the elementary solution of the radiative transfer equation Eq 323 Lecture 3 11s1 110eXp rs10leXp Ts1s mds For a non scattering medium in the thermodynamical equilibrium J B1T where BTis Plank s function Also for the nonscattering medium we have e ay1 kip where k is the mass absorption coefficient and p is the density see Lecture 3 Thus the solution of the equation radiative transfer for this case can be expressed as 11 s1 I lt0 expm s0 j exper sum31 ltTltsgtgtk pds 412 NOTE The optical depth in Eq412 is due to absorption only so 7109125 T etsds ilk1M5 4 Sun as an ener source Solar constant is total radiation emitted by the Sun reaching the top of the Earth s atmosphere Not a constant but it varies as a function of several parameters including sun activity sun spots distance between Sun and Earth 132 i39 1 3913 l 39f E1363 ACHIMI l E I v i 39 w wast I 1396 l l 39 39quot will L 1 A I v i 14 ty 39 is 39 i n In up ilr Yy39jly39 l Halt V quot51334quot was Jxrgmw n quot g m 1 i 1 l I l 2 I E i i Z i BTEI 3031 3233543556 37 EB 39 91191 92939495 96 939quot 9399 mm Figure 44 Measurements of solar constant from five independent spacebased radiometers since 1978 top have been combined to produce the composite solar irradiance bottom over two decades They show that the Sun s output uctuates during each 11year sunspot cycle changing by about 01 percent between maximums 1980 and 1990 and minimums 1987 and 1997 in magnetic actiVity The larger number of sunspots near the peak in the 11year cycle is accompanied by a rise in magnetic activity that creates an increase in solar radiation The capital letters are acronyms for the different satellite radiometers see L02 capture for figure 212 Lecture 8 Terrestrial infrared radiative processes Part 1Linebyline method Obiectives 1 Fundamentals of the thermal IR radiative transfer 2 Linebyline computations of radiative transfer in IR Required reading L02 422423 1 Fundamentals of the thermal IR radiative transfer Recall the equation of radiative transfer Eq 325 a b Lecture 3 for upward and downward intensities in the plane parallel atmosphere dITT T T T Ir Jr 325a d1i r 11 12 1ir Jir 325b and its solutions Eq326ab Lecture 3 13r lin exp 1 h TLT 326a jexp Jletmmdr Iir y Ii0 6Xp i 326b r r39 Jie y dr 1 r Z exp Infrared radiative transfer in the atmosphere Assuming non scattering medium in the local thermodynamical equilibrium the source function is given by the Plank s function E T see Lecture 4 and m lm kip where k is the mass absorption coefficient Assuming that the thermal infrared radiation from the earth s atmosphere is independent on the azimuthal angle p the equation of infrared radiative transfer for the monochromatic upward and downward intensities can be expressed as in the wavenumber domain allT r wy va dIl r Iiu m BAT and its solutions as 1 IJrIJrmexp Tr 1n ZJ CXP r T BVTr39 dr39 u Inn m 130 yexp gt 7 2quot ij exp BVTTd7 I o 8 la 81b 82a 82b To solve Eqs82a b for the entire atmosphere with total optical depth 15V two boundary conditions are required Surface assumed to be a blackbody in the IR emitting with the surface temperature Ts T IV Tw BAR BVTTV BAR Top of the atmosphere TOA 12 0 no downward thermal emission Ito m 0 Using the above boundary conditions the formal solutions for monochromatic upward and downward intensities are Mm Bvrquotexp Ti T I I 83a LICXp T T Bvr dr m m 1 J exp T T Bvr dr 83b ll 0 ll The formal solutions for monochromatic upward and downward intensities can be also expressed in terms of monochromatic transmittance By de nition the monochromatic transmittance is TV TV T eXp u and the differential form is off 139 1 TV M eXp dr Thus the formal solutions for monochromatic upward and downward intensities given by Eq83ab in terms of transmittance are Mm BvrquotTv 1quot w z r 84 I BVTJTVITTdT a elm 12mm d1 84b lierm I BM 2 Line bv line LBL quot of radiative transfer in IR LBL method is considered to be an exact computation of radiative transfer in the gaseous absorbingemitting inhomogeneous atmosphere accounting for all known gas absorption lines in the wavenumber range from 0 to about 23000 cm39l I nu Strategv to perform LBL ie to solve 17quot 9 9 M for the plane parallel atmosphere For a given wavenumber v For the j th atmospheric layerhomogeneous temperature T pressure pj length AZj For n th gas A Absorptlon coefficlent kwquot 1s L L CIJquot Zkvjnl 2 Sm 2f1JIJ 1 1 where l l L in the number of absorbing lines of nth gas at a selected v Sm and fml are the intensity and pro le ofthe lth line Optical depth tam of nth gas of jth layer rk mm mmjumj where M is the slant path for ngas in jth layer ie the amount of nth gas in jth layer Repeating above calculations for all gases n1 N we nd optical depth of j th layer N rm Z 1W n1 Repeating above calculations for all layers j1 J we nd optical depth of each layer Using calculated optical depth of each layer we nd the monochromatic upward and downward intensities from Eq83ab NOTE Similar strategy is used to solve Eq44ab via monochromatic transmission function for nth gas TVVl Tm eXp and using the multiplication law of transmittance we have the transmittance of jth layer of N absorbing gases T T T VJ v1 v2quot T vN o Multiplication law of transmittance states that when several gases absorb the monochromatic transmittance is a product of the monochromatic transmittances of individual gases T TT v12N v1 v2quot Tm 85 For many practical applications one needs to know not monochromatic intensity or ux but intensity or ux averaged over a given wavenumber interval Spectral intensity intensity averaged over a very narrow interval that BV is almost constant but the interval is large enough to consist of several absorption lines Narrow band intensity intensity averaged over a narrow band which includes a lot of lines Broad band intensity intensity averaged over a broad band e g over a whole longwave region We can de ne the spectral transmission function for a band of a width Av as 1 1 1 Mu Edev Eexplt rvgtdv Eexpekmdv 86 NOTE Spectral intensify requires the calculations of spectral transmission which requires the calculations of monochromatic optical depth which are done with LBL computations Spectral absorptance is de ned as 1 1 AV 1 Tvltugt Ea expmwv EJfI expekyuwv 87 LBL spectral resolution Because LBL computes each line of absorbing gases in a nonhomogeneous atmopshere the adequate selection of an integration step ie interval dv is required to calculate the spectral transmittance in the interval Av Av gt dv Because P decreases exponentially with altitude the line halfwidth and hence the integration step should be smaller at higher altitudes in the atmosphere Because of these variable resolutions the absorptions coefficients of two consequent layers must be merged 7 it is done by interpolating the coarserresolution of lower layers into the nerresolution of the higherlevel gt spectral absorptance for a given slant path is computed with the nest spectral resolution gt Continuum Absorption lines may have long wings eg depending on a line halfwidth To simplify calculations the wings of a line are cut at a given distance from the line center Thus the absorption coef cient of the line may be expressed as kv S 165 88 where kyc gives the absorption fraction in the wings called continuum absorption CKD Clou Kneiz s and Davies continuum model includes continuum absorption due to water vapor nitrogen oxygen carbon dioxide and ozone The water vapor continuum is based upon a water vapor monomer line shape formalism applied to all spectral regions from the microwave to the shortwave The absorption coef cient for the water vapor continuum is the sum of the self and foreign air broadening components 0111 k5 Um pw ivp01 89 as elf where pw and p denote the water vapor partial pressure and the air ambient pressure in atm respectively and cxelf and Fair are the self and airbroadening coef cients for water vapor In the 812 pm region the selfbroadening coef cient is parameterized as Roberts et al 1976 OWN2T a176XP3 V 810 where Tr 296 K a4 18 b5578 and b787x10393 for cmlfin cmz g39l atm39l And Smr knelf 0002 at Tr 296 K Moreover cxelf depends on as OldVJ URIV22 CXPCTr T 1 where c608 NOTE There are two competing effects on absorption of the water vapor continuum absorption coefficient increases as the temperature decreases but colder atmospheric conditions have less water vapor gt LBL numerical codes LBLRTM is developed in the ATMOSPHERIC AND ENVIRONMENTAL RESEARCH INC available to the scienti c community httpwwwrtwebaercom FASCODE FAST Atmospheric Signature CODE httpwwwvsafrlafmilProductLinesIRClutterfascodeast HJP Smith D Dube ME Gardner SA Clough FX Kneizys and LS Rothman FASCODE Fast Atmospheric Signature Code Spectral Transmittance and Radiance Air Force Geophysics Laboratory Technical Report AFGLTR780081 Hanscom AFB MA 1978 Lecture 3 The nature of electromagnetic radiation Obiectives 1 Basic introduction to the electromagnetic field gt De nitions gt Dual nature of electromagnetic radiation gt Electromagnetic spectrum 2 Main radiometric quantities energy ux and intensity 3 Concepts of extinction scattering absorption and emission 4 Polarization Stokes parameters Required reading S 21 221 222 23 24 41 Appendix 1 Additional reading Online tutorial Chapter 1 Sections 12 7 13 httpwwwccrsnrcangccaccrs1earntutorialsfundamchapter1chapter1717ehtm1 1 Basic introduction to electromagnetic eld Electromagnetic radiation is a form of transmitted energy Electromagnetic radiation is sonamed because it has electric and magnetic fields that simultaneously oscillate in planes mutually perpendicular to each other and to the direction of propagation through space gt Electromagnetic radiation has the dual nature its exhibits wave properties and particulate properties gt Wave nature nfrzdiatinn Ramaan canbe Lhnughl afasz mveling mnsvemwzve Elactric mm E my 31 A mm w a m gamma we ang unnng 2 ms The mm 2 Hummus 9 m5 mum m m w pm ma pupmcmu m m ammm Maman r ylning vecmr gves m aw af mam emxgy and the reman afpxapagnuan as m the cgs system uflxnts 51mm 3 herecxsmzypeedaflxgmmwcmxn 29979xl xm ZUUXIUXmsjmd 5n 5 vacuum pexmxmvny a elecmc mustard S is m nts af energy per ntume pu umt area 2 g Wm NOTE Ex means avecmrpmducl mm mm i is u en called inshntznzuus Puyming vectnr E ecause n ascdlates stupid mg a 6mm measuresxts mung mm ltsgt We same mm mural thaws a chmamm af the mm Wzvts are huacunzedbyfmuzncywzveknglh quad undying Fregueng is defined as the number of waves cycles per second that pass a given point in space symbolized by 7 Wavelength is the distance between two consecutive peaks or troughs in a wave symbolized by the 2 Wavelength lt gt Relation between 2 and 7 32 0 Since all types of electromagnetic radiation travel at the speed of light short wavelength radiation must have a high frequency 0 Unlike speed of light and wavelength which change as electromagnetic energy is propagated through media of different densities frequency remains constant and is therefore a more fundamental property Wavenumber is defined as a count of the number of wave crests or troughs in a given unit of length symbolized by v 33 UNITS Wavelength units length Angstrom A 1A lxlO3910 m Nanometer nm 1 nm1x10399 m Micrometer pm 1 pm lxlO39 m Wavenumber units inverse length often in cmquot NOTE Conversion from the wavelength to wavenumber 71 Vcml 100000m um 3 4 lm Frequency units unit cycles per second Us or s39l is called hertz abbreviated Hz Frequency gt Particulate nature of radiation Radiation can be also described in terms of particles of energy called photons The energy of a photon is given as Ephoton h V h ck hcv 35 where h is Plank s constant h 6625621103934 J s 0 Eq 35 relates energy of each photon of the radiation to the electromagnetic wave characteristics 17 and 7t 0 Photon has energy but it has no mass and no charge NOTE The quantized nature of light is most important when considering absorption and emission of electromagnetic radiation PROBLEM A light bulb of 100 W emits at 05 pm How many photons are emitted per second Solution Energy of one photon is Sphmon hch thus using that 100 W 100 Js the number of photons per second N is 100Js 1 1m7 100 x 05 x1076 hm Cmsil 66256 x10734 x 29979 x108 Ns 1 2517 x1020 NOTE Large number of photons is required because Plank s constant h is very small gt Spectrum of electromameljc radiation to energy or eqmvalenuy according to me Wavelength or frequency ENERGY mcxms ES 1cr5 1n 3 wavelength cm WAVELENGTH INCREASES Figure 32 The elemomagneuc specm THE ELECTROMAGNETIC SPECTRUM ankngllv m m m x m m w m I HT HTquot 107 W W In InH mquot m manH y r km a i 39 39 va W q 1 m Cry a s We W W Ennunnn mm nu We W173 satMY Mquot I0 W WV m N W N W H7 1 NYquot m 0 n7 Vnngyn W WW WWW war an m w w w vr w w m m39 1 m ultraviolel rays microwaves Vislble xrays gamma rays J i 400 nm 500 nm 600 nm 700nm Figure 33 Visible region omie electromagnetic spectrum NOTE 39 euui quot quot calledbytheir color eg blue green and read channels NF 39 quot quot 4 quot39 Lecture 5 V i 5 INWRED mmm quot1 i i c l IHANSMISSJDM w II inquot minimum mu W mamas m mow 5quot W m low Id Mn Mm 10m lDzM Mam mm mm Fimlro A quot 4 439 39 39 39 quot 39 m u r 39 39 39 uni uiii radiaiion A 39 39 39 39 doesn39t L interact much with air molecules and hence isn39t absorbed o In this course we study UV Visible infrared and microwave radiation Name of Wavelength Spectral equivalence spectral region pm Longwave Table 3 3 Frequency EXAMPLE Lband is used onboard American SEASAT and Japanese JERSl satellites 2 Basic radiometric uantities intensit and ux Solid angle is the angle subtended at the center of a sphere by an area on its surface numerically equal to the square of the radius 9 12 36 739 UNITS of a solid angle steradian sr r A differential solid angle can be expressed as 9 d9 2 sin 6d6d Iquot using that a differential area is ds r 619 r sin9 de EXAMPLE Solid angle of a unit sphere 411 EXAMPLE What is the solid angle of the Sun from the Earth if the distance from the Sun from the Earth is d15XlO8 km and Sun s radius is Rs 696XlO5 km R2 Q 676x10 5sr d Intensity 0139 radiance is defined as radiative energy in a given direction per unit time per unit wavelength or frequency range per unit solid angle per unit area perpendicular to the given direction I L 37 ds cos 6detdl I is referred to as the monochromatic intensity 0 Monochromatic does not mean at a single wavelengths 7 but in a very narrow infinitesimal range of wavelength A centered at 7 NOTE same name intensity specific intensity radiance UNITS from Eq37 J sec391 sr391 m392 um39l W sr391 m392 um39l ZetlliI Nari1 plq cleo sw 39zce 0 cos 6 Ears Figure 35 Intensity is the ow of radiative energy carried by a beam within the solid angle dQ a In general intensity is a function of the coordinates 17 direction 52 wavelength or frequency and time Thus it depends on seven independent variables three in space two in angle one in wavelength or frequency and one in time b In a transparent medium the intensity is constant along a ray 0 If intensity does not depend on the direction the electromagnetic field is said to be isotropic o If intensity does not depend on position the field is said to be homogeneous el UNITS from Eq38 J sec391 m392 um39l W m392 um39l From Eqs 3738 the ux is integral of normal component of radiance over some solid angle F j I cos6dQ 39 Q 0 Each detector measures electromagnetic radiation in a particular wavelength range M The intensity 1M and ux F M in this range are determined by integrating over the wavelength the monochromatic intensity and ux respectively 3 q 1M JIAdl39 FM IF1d 310 41 41 NOTE Many satellite sensors have a narrow viewing angle and hence measure the intensity not ux To measure the ux a sensor needs to have a wide viewing angle 3 The concepts of extinction scattering absorption and emission Electromagnetic radiation in the atmosphere interacts with gases aerosol particles and cloud particles 0 Extinction and emission are two main types of the interactions between an electromagnetic radiation eld and a medium eg the atmosphere General definition Extinction is a process that decreases the radiative intensity while emission increases it NOTE same name extinction attenuation Radiation is emitted by all bodies that have a temperature above absolute zero 0 K often referred to as thermal emission o Extinction is due to absorption and scattering Absorption is a process that removes the radiative energy from an electromagnetic eld and transfers it to other forms of energy Scattering is a process that does not remove energy from the radiation eld but may redirect it NOTE Scattering can be thought of as absorption of radiative energy followed by re emission back to the 39 quot field with quot quot 39 39 of energy Thus scattering can remove radiative energy of a light beam traveling in one direction but can be a source of radiative energy for the light beams traveling in other directions 0 Elastic scattering is the case when the scattered radiation has the same frequency as that of the incident field Inelastic Raman scattering results in scattered light with a frequency different from that of the incident light 4 Polarization Stokes parameters Polarization is a phenomenon peculiar to transverse waves 0 Electromagnetic radiation travels as transverse waves ie waves that vibrate in a direction perpendicular to their direction of propagation WOTF mlougll medla by altematlvely forclng the moleellles of ale medlllm elosel together then spreadlng them apart dlleeuon Hollzomal and vemeal polallzalloll are an example oflinear pnlarilztinn 55neos1lzemwn 311 wllele En ls the amplitude k ls ule plopagaaoll ol wave constant k 2111 m ls ule elleulalflequelley okc 2121 4p n ls the constant orlmtlal pllase 4p k e 0t4pnlsthephzse nnhewave Introducing complex variables Eq3 l 1 can be expressed as E E0exp z39qp 312 NOTE we use expiiq0 cosqp iisin The electric vector E may be decomposed into the parallel E1 and perpendicular Er components as We can express El and E in the form E1 E0 cos k2 wt p10 E E0 cos k2 wt pro Then we have El El0 cos cos 110 sin sin lo E E0 coscoswo sin sinltoo where g kz at Performing simple mathematical manipulation we obtain ElE102 E E02 2E1E10EE0cosAgp sin 2qu 313 where A p p 0 p 0 called the phase shift Eq3 13 de nes an ellipse gt elliptically polarized wave Ifthe phase shift A a n 7 n0 1 2 then sin Aw 0 andcos Aw i1 and Eq 313 becomes Er E10 ErO 2 E J0 or E i WE 314 Eq3 14 de nes straight lines gt linearly polarized wave Ifthe phase shift A a n 1 2 n 1 3 and E10 Era Eg then sin Aw i1 and cos Ago 0 and Eq313 becomes Ef E E02 315 Eq315 de nes a circle gt circular polarized wave NOTE The sign of the phase shift gives handedness righthanded and lefthanded polarization Unpolarized radiation or randomly polarized is an electromagnetic wave in which the orientation of the electrical vector changes randomly If there is a de nite relation of phases between different scatterers gt radiation is called coherent If there is no relations in phase shift gt light is called incoherent 0 Natural light is incoherent 0 Natural light is unpolarized o The state of polarization is completely defined by the four parameters two amplitudes the magnitude and the sign of the phase shift see Eq 3 13 Because the phase difference is hard to measure the alternative description called a Stokes vector is often used Stokes Vector consists of four parameters called Stokes parameters intensity I the degree of polarization Q the plane of polarization U the ellipticity V Notation ms or IQUV V o Stokes parameters are de ned via the intensities which can be measured I total intensity Q 10190 differences in intensities between horizontal and vertical linearly polarized components U L45 7145 differences in intensities between linearly polarized components oriented at 450 and 450 V Irclillcr differences in intensities between right and left circular polarized components 0 Stokes parameters can be expressed via the amplitudes and the phase shift of the parallel and perpendicular components of the electric field vector I E 30 E I Q E2 EZ 316 U ZEMEIO cosA V 2EroElo EXAMPLE Stokes parameters for the vertical polarization For this case El 0 E 1 U E30 E1 0 0 V 0 0 Lecture 10 WW Absorption band models Obiectives 1 Concept of the equivalent width Limits of the strong and weak lines 2 Absorptionband models Regular Elsasser band model and Statistical Goody band model 3 CurtisGodson Approximation for the inhomogeneous path Required reading L02 44 1 Concept of the 39 39 width Limits of the strong and weak lines First let s consider a homogeneous atmospheric layer ie the spectral absorption coefficient k does not depend on path length Recall Lecture 8 where we have de ned the spectral transmission function for a band ofawidthAvas T LO LIBX k udvLJeX SfV V udV 7 AVAV p v AVAV p 0 and spectral absorptance Av u 1 TV u Lja exp kudv AV AV Equivalent width is de ned as WW AVAV lil GXPHCVWldV 101 1 where W IS in units of wavenumber cm o The equivalent width is the width of a fully absorbing Al rectangularshape line Spccrnl Alvsomtancc Figure 101 Schematic illustration of the equivalent width The dotted rectangular area is equal euer y Eguivalentwidth 0 Lorentz zo le Using kv S v 7 v0 and the Lorentz pro le of a line we have 1 S AA E I 1 exp jd 102 AV 0 This integral can be expressed in term of the Ladendurg and Reiche function Lx as W A V 27mcLx 103 where x Sta2m S is the line intensity and u is the absorber amount NOTE The Ladendurg and Reiche function Lx in Eq103 is given by the modi ed Bessel functions ofthe rst kind of order n Lx x exp7gtc1o x Igtc where 7 n momma 21 d J xt7 Icosm expix cosn d 0 For small Lx is linear with its asymptotic expansion Lx x1 For large in Lx is proportional to a square root ofx Lx Zxnm1 Case of weak line absorption either kV or u is small gt k u ltlt1 Using the asymptotic of LX for small X we have Su Su AV u 2 AK 27raLxAV 2 27m W A V 7m V V Thus Su AV u E is called the Linear absorption law 104 Case at strong line absorption SuIra gtgt1 Using the asymptotic of LX for large X we have Avu 27raLxAV 27239a12 xAV AV 7239 27239a 2S AV2 SuaAV 7239 2 7m Thus V Su 05 AVQquot 2 T s called Square root absorption law 105 2 Absorption band models Band is a spectral interval of a width Av which is small enough to utilize a mean value of the Plank function B V T but large enough so it consists of several absorption lines Let s consider a band with several lines Two main cases can be identi ed 1 lines have regular positions 2 lines have random positions Two main types of band models regular band model and random band models Re lar Elsasser band model consists of an in nite array of Lorentz lines of equal intensity spaced at equal intervals Example This type ofbands is similar to P and Q branches oflinear molecules For example the spectrum oszO in 778 pm band39 the spectrum ofCOz in 15 um band The absorption coefficient of the Elsasser bands is S or k 106 V 21 vin 392zx2 where 5 is the line spacing ie the distance in wavenumber domain cm39l between the centers of two nearest lines Figure 102 Schematic depiction of the absorption coef cient in the Elsasser regular band model for three different values of y xS NOTE The parameter ofy xS can be regarded as a grayness parameter ify is large then adjacent lines strongly overlap so that line structure is increasingly obscured39 for small y the lines are well separated Using Eq106 one can calculate the spectral absorptance as see derivation in L02 pp139141 A7 erf 107 17rS 0m 6 where erf x Jexp x2 dx Values of erfx are available from standard 7139 0 mathematical tables Principle of 39 39 39 random models Many spectral bands have random line positions To approximate this type of bands various statistical models have been developed Example H20 63 pm vibrationalrotational band and H20 rotational band are characterized by random line positions Assumgtitms n randomly spaced lines with the mean distance 5 so that Av n5 lines are independent and have identical shapes probability density of the strength of i th line is pS DifferentpS give different models for instance Goody Malkmus etc Strategy derive mean transmission by multiplying transmission of each line at a particular v and integrating over probability distributions of line positions vi and line strength S for each line TV HJ dvlJ dvn I pSleXp uSlfV V071d91 A V Av Av 0 co J pltSgtexplt uSfltv vO dS n L dVi3 PSieXP uSifVV0iaSi AVAV 0 y i1 NOTE Above equation uses that if lines in a band are uncorrelated the multiplication law see Lecture 8 works for average transmittance Tv12 TV1TV2 Since in the above equation all integrals alike we have 1 0 TV ml 51v pSeXp quvdSquot 00 108 1 VI 1 El 51v pSl eXp qu vdS The mean equivalent width may be de ned as W 1 MS I leXpqu ltvgt1dvdS 109 0 Av Recalling that Av n5 Eq108 can be rewritten in terms of the mean equivalent width giving the mean transmission as 1 n L 1 V quotan 1010 x n Since llmngtw1 gt eXp X we have n NOTE Single line transmission is l WAv but for many random lines it is exponential in the mean equivalent width Statistical Goody band model Consider a band consisting of randomly distributed Lorentz lines Assuming that the probability distribution of intensities is the Poisson distribution MS 5 1 eXPSS 1012 where the g in the mean intensity S SpSdS O39u8 For the Lorentz pro le with the mean halfwidth x the spectral transmittance can be expressed as 712 14 14 TV 2 6X10 1E 1013 Thus Eq10 13 gives the mean spectral transmittance for the Goody random model as a E 5 function of path length n and two parameters Malkmus model has a higher probability of weak lines assumes that the probability distribution of intensities is pS S4 exp S and for a Lorentz line shape the mean transmittance is 4g 12 720 Ll TV eXp 2 1 1 1014 Weak line limit Su For ltlt 1 Eq1013 g1ves 7m T7 exp 5 1015 5 Strong line limit Su For gtgt 1 Eqs10l3 and 1014 give 705 7W 1016 TV exp 6 1 Band Transmission Models I I 0 oo 0 O 04 Band Transmittance 0 N nu 01 l 10 100 1000 55 H Figure 103 Comparison ofthe Elsasser solid and randomMalkmus dashed band models for several Values of y tot5 labeled on cmves For both models curved approach the gray limit Beer s law When y gtgt 1 3 quot F I39 39 39 for39 39 Dath All discussion above Was for homogeneous path because band parameters are for one pressure and temperature In real atmosphere of Varying T and P some adjustmenw of the band models are needed to account for the inhomogeneous path 1 JkpuTu du Stream reduce the radiative transfer problem to that of homogeneous path With some sort ofavemged Values ofuquot Tquot and pquot so that optical depth can be computed accurately Lecture 5 AW Basic properties of gases aerosols and clouds that are important for radiative transfer modeling Obiectiver 1 Structure of the Earth s atmosphere and radiative transfer modeling 2 Atmospheric gases 3 Aerosols 4 Clouds 5 Refractive indices of common atmospheric particulates Rem edrewg39 L02 31 51 pp169176 1 Qtrllrtllre ofthe r mm 39 39 Inuueling Propagation of the electromagnetic radiation in the atmosphere is affected by its nre Ilre an Radiative transfer in the atmosphere in NWP and GCMs is commonly solved in one dimension based on the concept ofthe plane parallel atmosphere see gure 34 in Lecture 3 EXAMPLE In global and regional models radiative transfer is solved in each column de ned by the model grid size The vertical resolution is the same as the number of vertical layers in the model Threedimensional radiative transfer codes are mainly used in cloud research and in LES models and vegetation studies quotI 39ll 1 I I ll H Ih39p 1 39 a 4 4 v 39 39 39 mmin quot 39 madelingeachmodelinCludeS pro les of T P and concentration of main gases 70 39 80 39 39 Mescsphere 50 l W Stratosphzre g 40 7 7 393 3 sldl 76 I Lrop subtropsum sublrapwm 20 i subarcwin 10 w 7 Troposphere l l n 90 200 210 220 230 240 250 260 270 250 290 300 Temperature K Figure 51 Temperature pro les of the standard atmospheric models o en used in radiative transfer calculations Standard US 1976 atmosphere is representative of the global mean atmospheric conditions Tropical atmosphere is for latitudes lt 30 Subtropical atmosphere is for latitudes between 300 and 45 Subarctic atmosphere is for latitudes between 450 and 60 and Arctic atmosphere is for latitudes gt 60 Interactions of ii 39 with radiation Gases 7 all scatter radiation and some can absorb and emit radiation depending on their molecular structure Particulates aerosol and clouds 7 all scatter radiation and some can absorb and emit radiation depending on their refractive indices determined by composition 2 Atmospheric gases Table 51 Gaseous composition of the Earth s atmosphere gases Oxygen 0 20948 Mixed in most of the middle atmosphere in the tropics km dissociation 000001minimum 00365 up to 100 000006 decreases slightly with height in the 000003 at dissociates in stratosphere and 0000009 altitude km amp ON 0 O N o l l l 10 9 10quoti 10F1 10 10 10 mixing ratio 0 l 101 1010 Figure 52 Vertical pro les of mixing ratios of some gases in the atmosphere The amount at the gas may be exgressed in several ways i Molecular number density molecular number concentration molecules per unit volume of air ii Density molecular mass concentration mass of gas molecules per unit volume of air iii Mixing ratios Volume mixing ratio is the number of gas molecules in a given volume to the total number of all gases in that volume when multiplied by 106 in ppmv parts per million by volume Mass mixing ratio is the mass of gas molecules in a given volume to the total mass of all gases in that volume when multiplied by 106 in ppmm parts per million by mass NOTE Commonly used mixing fraction one part per million 1 ppm 1X10396 one part per billion 1 ppb 1X10399 one part per trillion 1 ppt 1X1 039 iv Mole fraction is the ratio of the number of moles of a given component in a mixture to the total number of moles in the mixture NOTE mole fraction is equivalent to the volume fraction NOTE The equation of state can be Written in several forms using molar concentration ofa gas c uv P c T R using number concentration ofa gas N c NA P N T RNA or P N T k5 using mass concentration ofa gas q c mg P q T R mg The mucture 0f 39 39 39 their em V forms and their 1111M to absorb munultimo Permanent Elem m Dipole Momenr Mulewe Sautmre linear N OWE m magnetir dipole Niuogen M linear No CnibonMonoxide 6H linear Yes Carbon Dioxide SD H lineal No Niuoumxide l a N lineal Ves o Watev asymmetrlr top Ves H H ozone 7 asymmemr top Yes Q r 62 Merlvane spheviral mp Yes Figure 53 Molecular structures of key atmospheric gases and the dipole moment status Linear molecules C02 N20 Csz all diatomic molecules Symmetric top molecules NH3 CH3CL CF3CL Spherical symmetric top molecules CH4 xxxx Asymmetric top molecules H20 03 o The structure of a molecule determines whether the molecule has a permanent dipole or may acquire the dipole The presence of the dipole is required for absorptionemission processes by the molecules see Lecture 6 3 Aerosols Atmospheric aerosols are solid or liquid particles or both suspended in air with diameters between about 0002 pm to about 100 pm 0 Interaction of the particulate matter aerosols and clouds particles with electromagnetic radiation is controlled by the particle size composition mixing state shape and amount 0 Atmospheric particles vary greatly in sources production mechanisms sizes shapes chemical composition amount distribution in space and time and how long they survive in the atmosphere ie lifetime Primary and secondary aerosols Primary atmospheric aerosols are particulates that emitted directly into the atmosphere for instance seasalt mineral aerosols or dust volcanic dust smoke and soot some organics Secondary atmospheric aerosols are particulates that formed in the atmosphere by gas toparticles conversion processes for instance sulfates nitrates some organics Location in the atmosphere stratospheric and tropospheric aerosols Geographical location marine continental rural industrial polar desert aerosols etc Spatial distribution Atmospheric aerosols exhibit complex heterogeneous distributions both spatially and temporally Anthm 2 man Mzmtdnammlazmxa proeesses agneulture39s aetavrties Natural Salutes vanoussearsa1tonststorm nomass burning voleanie debris gas to pamcle eonversion atemu al camgaxman Individual eluernieal species sullate sofa nitrate Nos sum elemental earbon searsalt Nacl minerals e g quartz 5103 M I internally mixed partieles Parade x52 3 tumour w A 7 a if y nndcv sa nn w 39 t r r w a lag v w m gamut diam5m r e it new H 7 4 arhr PS e hamwm ulna a rtase lemma 7 a 7 Wu F Figure 5AA classlcal Viewquot ofthe distnbutlon ofpamcle mass of atrnosplnene aerosols from Whitby and Cantrell 1976 NOTE the nuelei rnoole about 0 005 urn lt d lt 0 1 pm and aeeurnulation mode01pm lt d lt 2 5 pm o The particle size distribution of aerosols are commonly approximated by the analytical functions such as lognormal power law or gamma function Lognormal function No lnrr 2 N if r J5 lno reXP 211162 Normalization j NrdrN0 52 Ifthree size modes are present eg see Figure 54 then one takes a sum ofthree log normal functions N lnr r0 2 NU ex 7 quot 53 Z 427139 lno x r P 21n039x2 where Nr is the particle number concentration Ni is the total particle number concentration of ith size mode with its median radius r0 and geometric standard deviation 0 NOTE Surface area or volume mass size distributions can be found using the k moment of the lognormal distribution k2 or k3 respectively I rkNrdr Norok eXpk2ln 0392 2 54 5 Clouds Major characteristics are cloud type cloud coverage and distribution liquid water content of cloud cloud droplet concentration and cloud droplet size Cloud droplet sizes vary from a few micrometers to 100 micrometers with average diameter in 10 to 20 um range Cloud droplet concentration varies from about 10 cm393 to 1000 cm393 with average droplet concentration of a few hundred cm393 The liguid water content of typical clouds often abbreviated LWC varies from approximately 005 to 3 gwater m393 with most of the observed values in the 01 to 03 gwater m393 region NOTE Clouds cover approximately 60 of the Earth s surface Average global coverage over the oceans is about 65 and over the land is about 52 Table 52 Types and properties of clouds Height of Freq over Coverage Freq over Coverage base km oceans over oceans land over land Type W W Low level StratocumulusSc 02 45 27 18 Stratus St 02 ScSt ScSt ScSt ScSt Nimbostratus Ns 04 6 6 6 5 Mid level Altocumulus Ac 27 46 22 35 21 Altostratus As 27 AcAs AcAs AcAs AcAs High level Cirrus Ci 718 37 13 47 23 Cirrostratus Cs 718 CiCsCc CiCsCc CiCsCc CiCsCc Cirrocumulus Cc 718 Clouds with vertical development Cumulus Cu 03 14 5 CumulonimbusCb 03 10 6 7 4 0 Cloud droplets size distribution is often approximated by a modi ed gamma distribution Nr may 5395 n ail No L eXp rrn r Where No is the total number of droplets cm393 1 n in the radius that characterizes the distribution a in the variance of the distribution and 1quot is the gamma function Table 53 Characteristics of representative size distributions of some clouds for 1 2 Cloud type No rm rmax re 1 011139s Em mm mm g In393 Stratus over ocean 50 10 15 17 0105 over land 300400 6 15 10 0105 Fair weather cumulus 300400 4 15 67 03 Maritime cumulus 50 15 20 25 05 Cumulonimbus 70 20 100 33 2 5 Altostratus 200400 5 15 8 06 Mean radius rm a 1 1 n Effective radius re a 3 1 n o rot man 39 quot 39 L 39 39 eloud ale parameterized as afunction ofthe e 39ecu39ve radius and liquid water content LWC The eiiective radius is de ned as I r3Nrdr 7 56 I r2Nrdr whereNr is the droplet size distribution The liquid water content LWC is defined as 4 3 LWCpWV prz7 Nrdr 57 CldeuC staLv lce crystals present in clouds found in the atmosphere are often sixsided However there are variations in shape plates nearly at hexagon columns elongated at bottoms s 4 39 A Aquot 39 snow ake shape lce crystal shapes depend on temperature and relative humidity Also crystal shapes t t l l can 1 1 u 5 DE 195 FapHcan mm V 3925 NW laal REDYIumr Vrnhlz no a r i win 5 Refractive indices of common 39 in particulates Refractive index or optical constants mn ik is the material properties of dielectric that determines its radiative properties In general each material has its own spectral refractive index The imaginary part k of the refractive index determines the absorption of the wave as it propagates through the medium the real part n of the refractive index gives the phase velocity of propagation V It is believed that the refractive indices of the bulk material apply down to the smallest atmospheric aerosol particles V The refractive index is a function of wavelength Each substance has a specific spectrum of the refractive index V Particles of different sizes shapes and indices of refraction will have different scattering and absorbing properties Examples 0 re active indices a Index of Refraction of Water and Ice Real Part TB I II I 7 Water IOU C Ice 5U C quotquot I I I I W I O I 01502 03 05 07 I 15 2 3 4 5 7 10 IS 20 30 50 WaveIengIh um b Index of Refraction ofWater and Ice Imag Pan l l I I I 01 001 0 OOI 0000I r r r Ie 05 r r r I I u 1 I I IeOB 5 07 I Water 10 CI Ie OB r 3950 C 1809 I II 01 01502 03 05 07 I 15 2 3 4 5 7 10 15 20 30 50 WaveIength um Figure 56 The complex refractive index of water and ice IMAGINARY mm D nzmncnv anx we x um 1 H wnz Figure 57A classical pmquot shnwmg the imagnary pan nthe re acnve 1ndEXes nf snme aemsnl materials Buhren and Huf nan Fig 5 16 NnTF Rm nverall absnrpnnn x e absmpunn cmef ment 15 as nntmlled by pamde sue 1 Covmrl39omu Hapl dmg in I M quotTIME BaH In qunL ylLL Wx kgr gM EWS 4 WVNW C 15er 77 foomu 39o from W 121M C0 6 L f lIM 1 IL 4k ta 15 a Wm M 3L L 7t If z 1L7 f ygtt 0 quot tvlv quot051 auhrcyLsxi 11 ZLUL 14141 lt y gt I M quot71 1 a 5 WC Lid Lag In My gJWAL CAIC Via 441 rtwrnlg M QLVIM fin 4 WI uxivg Mm r x Mv39faGOM oz Ly z g g zyt 14x1 Muhr 4 tk4 ILL only 4xV MMffk 5quot yt39l 5 law Nxa W for as WI MN our 4 1405141 far I 3 c Mer l e7 Solunimx of Ig 4 5fgt g ITgt TA L7 7qu If 7 Law Axxf 11 gtF d4quAl u 46 a Caln IQ de fami ngfb 11141115 of if s Law1 4 7 1ny1gt 2L9 L y 43 my Uh ltgz gnygt in AC 20 J If yaw MHM 0 quot y 39U5quotl7 a 7 LV awfaoovoufawga 39fdno 39sw 1 th Wt have T r F X 4 5 7 z 1 gt g M fwm 67 clcu raw 6171 mgdw l Lag My gtH 69 a 39 0 2L0 39 39 7L tau1 m 0 LM Lecture 24 Radiative and radiativeconvective equilibrium Obiectives l Radiative equilibrium 2 Radiative and radiativeconvective equilibrium models Appendix Derivation of the Eddington gray radiative equilibrium Required reading L02 83 Additional reading Goody RM and YL Yung Atmospheric radiation Theoretical basis Chapter 9 Oxford Univ Press 1989 1 Radiative equilibrium In a real atmosphere solar heating rates do not equal to IR cooling rates This imbalance is the key driver of atmospheric dynamics Let s consider a hypothetical motionless atmosphere radiative transfer processes only Then the climate state temperature pro le is determined by the radiative equilibrium The radiative equilibrium climate model is a model predicting the atmosphere temperature pro le of an atmosphere in rad1at1ve equ111br1um 6 0 z Under the gray atmosphere assumption we can solve for the temperature pro le analytically Mi gray radiative equilibrium results see Appendix for the full derivation Assumptions dF 1 Radiative equilibrium 0 0 2 2 Gray atmosphere in longwave 3 No scattering and black surface in longwave 4 No solar absorption in the atmosphere 5 Eddington approximation I u I 0 11 Longwave ux pro le Flust 1 and FlrFmr 241 where F 1 770 4 Atmosphere blackbody emission and temperature pro les F lr and T4rT24r 242 31 2 27239 Surface temperature is discontinuous with the atmosphere hotter FSMVL 27239 BS Br and T T1r 243 Imglications Greenhouse effect 7 larger 1 increases surface temperature Runaway greenhouse effect 1 increases gt TS increases Positive feedback 7 higher temperature gt greenhouse gases Eddington gm radiative equilibrium temgeratures If one wants to have the temperature profile in terms of height one needs to relate optical depth to height Assume that an absorber has the exponential profile pa p0 eXpZ H a 244 So the profile of optical depth is 12 I j pa 2612 I pOHa eXp zHa r eXp zHa 245 Temperature pro le T4z T4 lrexp zHu 246 Lapse rate d Tz 2exp zHa 247 dz 8 1 5 T Ha Implications Low 1 gt stable atmosphere Smaller scale height Ha of the absorber causes steeper lapse rate Steepest lapse rate near the surface 20 2 quot quot and rm39h n quot equilibrium models 0 Radiative equilibrium climate models solve for the vertical pro le of temperature using accurate broadband radiative transfer models 0 Model inputs vertical pro le of gases aerosols and clouds Iterates the dF temperature pro le to arch1ve equ111br1um 1e zero heat1ng rates or 6 0 z 0 Climate feedbacks can be included by having water vapor surface albedo clouds etc depend on temperature Solvin 0r radiative e uilibrium 6T Iterate the temperature pro le Tz to get zero heat1ng rates 6 0 t 1 Time marching method T at tl time step from heating rate at time t T 1zk Tz At 248 2 Dich salvz time marching Radmtwe equxlxbnum temperature pro les Show See Fxgme 241 below coz may atmosphexe has less steep pro le Earth s stratosphere warms due to UV absorption by ozone Most greenhouse effect from water Vapor 23 HgOLS Hzoecogrusu f C02 us i 10 3 E 33 m 2 ll 391 D C a a m 7 E 100 a 1000 TEMPERATURE m Finlro 14 I quot 39 39 39 39 39 gases in a clear sky at 35 N in April LS means he me effects ofboch longwave and shortwave radiation are included from Manabe and Strickler 1954 Reruh the rad1at1ve equ111br1urr1 urface temperature 1 the h1gh and the temperature pref11e1ur1real1t1c 1H mmw eeeu threheldfercer1vect1er1 Fix aurr1e cenvectlen 11m1t 1ape rate to ltyceg65Kkrr1 1 Move heat 11ke cenvectmn 1fyE exceeded adjutterr1perature 0 ya ach1eved and heat 1 cenerved 2 Parameter1zecer1vect1ve uxe g 1 219 Reruh 0f the RCE model developed by Mamabe 11ml Striekkr 1964 2 440 10 t rao 3 E 1 3 V j J 4 n 3 20 E U P m m 100 PURE RAD1ATVEEQUIL E DRY ADIAEATIC ADJ 3 65 Ckm ADJ 10 1000 0 1115111111111 180 220 260 300 340 TEMPERATURE K mm 711 1 1 temnerature pref1 e for two value of ya furclear sky 23Illll IIIIIIILILIILIIlI f i rquot 40 t 1 x 10 i ll 30 a I v 3 m 1 5 P5 1 a 20 g g t R E 100 H20 3 3 H20c02 1 H20COZO3 l uo 100 llllllllllll l l lTll lIll ll 0 100 140 180 220 260 300 TEMPERATURE K gure 241D quot39 quot Jul variou atmospheric gases in a clear sky at 35 N in April Cam in ures 261and263 J 439 39 quot39L 39 39 fairlv aLLL aLCJUI J L Llatitudinal and seasonal dependence is not correct f nr quot Aggendix Derivation of the Eddington gray radiative equilibrium Assumptions 6 V dFm Rad1at1ve equ111br1um d 0 Z 7 Gray atmosphere in longwave 8 No scattering and black surface in longwave 9 No solar absorption in the atmosphere 10 Eddington approximation Iu 10 11y Since the atmosphere is gray all wavelength are equivalent one can write the wavelength integrated thermal emission radiative transfer equation no scattering u I B dr where I is the integrated radiance W In392 st39l 1 increases downward and p gt0 in the upward direction Note that deriving the variation of B with the optical depth 1 is equivalent to determining the temperature pro les since the blackbody emission is a function of temperature only Using the Eddington approximation the net uX positive upward becomes 1 47239 Fm 2 1M1 T11 71 The radiative equilibrium assumption implies that Fm and II is constant with optical depth Integrating the above radiative transfer equation over dp gives d 1 1 1 27rd J Lady 27239 Idy 27239 Edy T 1 71 71 dF M 4710 4729 d1quot Under the radiative equilibrium assumption we have 103 Integrating the radiative transfer equation over udp gives d 1 1 1 27r J Iyzdy 27239 Iydy 27239 Bydy d7 71 71 71 Since B is isotropic the last term drops out leaving 47239 d 47239 11 3 dr 3 dB 1 d 139 Thus the solution for B is simply a linear function of optical depth B L39 30 111 Constants B0 and 11 need to be determined from the boundary conditions Top of the atmosphere First boundary condition no thermal downwelling uX 0 27239 FRO 27 Indy 72290 11 0 71 so we have 3 11 330 or Fm 27280 Second boundary condition upwelling longwave ux is equal to the absorbed solar uX Fsun 1 27239 FRO 27239 j udy 7280 11 Fm 0 Recall that the absorbed solar uX Fsun is F 51m 1 7F0 4 3 Puttlng 1n 11 EB0 glves Fm 271790 Fm So now we have the B0 and 11 and thus the atmosphere Planck function pro le is determined FM 1 2139 31 27239 2 The nal step is to apply the boundary condition at the surface to obtain the surface temperature Ts This boundary condition is that the emitted uX by the surface equals to the sum of the downwelling shortwave and longwave ux at the black surface Lecture 6 Basics of gaseous absorptionemission Line shapes Obiectives 1 Basics of atomic and molecular absorptionemission spectra 2 Spectral line shapes Lorentz pro le Doppler pro le and Voigt pro le Required reading L02 13 Recommendedadvanced reading McCartney EJ Absorption and emission by atmospheric gases John WileyampSons 1983 I u I 1 Basics of atomic and absoi spectra I Atomic abam spectra 0 Radiation emission absorption occurs only when an electron makes a transition from one state with energy Ek to a state with lower higher energy for emission Ek Ej hCV Figure 6 1 AbsorptionEmission processes Bohr s model of a hydrogen atom o The energy level is given as Equot 2 n123 61 where RH is the Ryberg constant l092X105 cm391 for hydrogen h is the Planck s constant and c is the speed of light The Wavenumber uf emlsslunabsurphun llnes ufhydmgen elem 6 2 wherej and k are lntegers de mng me luwer and hlgher energy levels respectlvely lnlrared Ullravwlel Lyman Balmer senes E4 4135 ev Paschen E3 4 V senes 2 E264DEV E 436W Figure l Energy level dlagzm fur the hydmgen elem Black Early Figure 63 Examples of awmxc 5p ecua gt Molecular AbsorptionquotEmission Spectra spectroscopy and quantum theory Three types or absorptionemission spectra 0 Sharp lines of Emma man u Aggregauons genes oflmes called bands m Spectral continuum mendmg over a broad mnge ofwavelengxhs Line 1 Band Continuous spectra wavelength gt Figure 64 Concept of a line band and continuous spectra Main Jrlvin nhvsicalv 39 39 0f 39 39 39 1 The origins of absorptionemission lie in exchanges of energy between gas molecules and electromagnetic eld 2 In general total energy of a molecule can be given as E Erm Evib Eel Etr Em is the kinetic energy of rotation energy of the rotation of a molecule as a unit body about 1500 cm391 farinfrared to microwave region Evil is the kinetic energy of Vibration energy of Vibrating nuclei about their equilibrium positions about 500 to 104 cm391 near to farIR Eel is the electronic energy potential energy of electron arrangement about 104105 cm391 UV and Visible Etr is translation energy exchange of kinetic energy between the molecules during collisions about 400 cm391 for T 300 K 0 From Emlt Etr lt Eviblt Eel follows that i Rotational energy change will accompany a Vibrational transition Therefore Vibration rotation bands are often formed ii Kinetic collision by changing the translation energy in uence rotational levels strongly Vibrational levels slightly and electronic levels scarcely at all 0 Energy Em Evib and Eel are quantized and have only discrete values speci ed by one or more quantum numbers see below Not all transitions between quantized energy level are allowed they are subject to selection rules 3 Radiative transitions of purely rotational energy require that a molecule possess a permanent electrical or magnetic dipole moment NOTE A dipole is represented by centers of positive and negative charges Q separated by a distance d the dipole moment Q d Table 61 Atmospheric molecule structure and dipole moment status see also Table 53 Molecule Structure Permanent May acquire dipole moment dipole moment N2 N H N N0 N0 02 O O N0 N0 CO C 0 Yes Yes C02 0 C O No Yes in two vibrational modes CH4 H H No Yes 1 in two vibrational modes H H NOTE If charges are distributed symmetrically gt no permanent dipole moment gt no radiative activity in the farinfrared ie no transitions in rotational energy Example homonuclear diatomic molecules N2 02 NOTE CO2 and CH4 don t have permanent dipole moment gt no pure rotational transitions But they can acquire the oscillating dipole moments in their Vibrational modes gt have Vibrationrotation bands NOTE CO N2O H20 and 03 exhibit pure rotational spectra 4 Radiative transitions of vibrational energy require a change in the dipole moment ie oscillating moment Figure 65 Vibrational modes of diatomic and triatomic atmospheric molecules N N 2 no vibrational transition 02 symmetric stretching mode a uuuu C0 single vibrational mode C 02 gt V1 symmetric stretching mode f gt radiatively inactive 9 9 V2a two bending modes have same energy 6 V2 degenerated modes V3 asymmetric stretching mode gt radiatively active NOTE Homonuclear diatomic molecules N2 and 02 don t have neither rotational nor vibrational transitions because of their symmetrical structures gt no radiative activity in the infrared But these molecules become radiatively active in UV NOTE The number of independent vibrational modes called normal modes of a molecule with Ngt2 atoms are 3N6 for nonlinear molecules and 3N5 for a linear molecule NOTE Both H20 and 03 have three normal band v1 v and V3 all are optically active NOTE CH4 has nine normal modes but only V3 and v4 are active in IR 5 Rotational Vibrational transitions Pnre rotational transitions can be understood by evoking the notion of a rigid quantized rotator Linear diatomic 02 C0 N0 Linear triatomic C02 N20 2 degrees of rotational freedom and 2 equal moments of inertia gsymmetric top bent triatomic 03 H20 3 degrees of rotational freedom and 3 unequal moments of inertia Figure 66 Axes of rotational freedom for linear and asymmetric top molecules molecules Let s consider a diatomic molecule with masses m1 and m at distances r1 and r from their common center of gravity The moment of inertia of this twomass rigid rotator is 2 2 1m m1r1 m2r 2 If 1 is the distance between the atoms we have m2 m1 1 r and r2 r m1m2 m1m2 Introducing the reduced mass m as mm my 1 2 m1m2 mm 1 2 2 y 2 wehave I 7 mr m1m2 The angular momentum L of a rigid rotator is de ned as 2 L I ma 2 m39r a where n is the angular velocity rad sec39l The kinetic energy of a rotator is equal 1 1 EM La Ima2 2 2 Classic rotator both angular momentum and rotational kinetic energy are continues Quantized rotator Quantum restrictions on rotational energy as a consequence of the quantum restrictions on angular momentum which found from a solution of the Schroedinger equation The quantum restrictions on angular momentum are I h 12 m0 JJ 1 27139 J is the rotational quantum number J 0 1 2 3 h is the Planck s constant Thus we have 01 as B is the rotational constant and it depends on the moments of inertia Im of a given molecule as h 8720quot Units of B LENGTH391 Selection rules A 1 for absorption and A 1 for emission Consider rotational transition between the upper energy level E and lower energy level E We have for upper level E B h cJ J 1 and for lower level E B h cJ J 1 Thus AEjthJ 1thJ Recalling that Vh C position of a pure rotational line is given by mquot gt equally spaced lines because B is constant for a given molecule NOTE The nonlinear molecules H20 and O3 asymmetric tops with three moments of inertia give very complex spectra Pure Vibrational ener Similar to the derivation above one can introduce a classical vibrator whose energies are continuous and then apply the quantum restrictions from the Schroedinger equation The allowed energy levels are Evk h c vk vk 12 64 where Vk is the wavenumber of the k normal vibrational mode Vk is the Vibrational quantum number Vk 01 23 For pure Vibrational transition we have AE h Vk Combine vibrational rotational energy B h c J J1 h c vk vk 12 65 0 Because Evil gt Em the spectrum of the combined transitions is a series of rotational lines grouped around the vibrational wavenumber ForAI1wehavev Vk ZBJ J 123 For AI 1 we have v Vk ZB J 1 J 01 2 3 where J is the rotational quantum number in the excited vibrational state Vk INTERUPTED VERTICAL SCALE V Wavenumber v Figure 66 Simultaneous transitions in Vibrational and rotational energies denotes lower energy level and denotes upper energy level NOTE P branch is the rotational transitions following the selection rule AJ 1 Q branch is the rotational transitions following the selection rule AJ 0 R branch is the rotational transitions following the selection rule AJ 1 Equot Spectral line shapes Lorentz pro le Doppler pro le and Voigt pro le Three main properties that de ne an absorption line central position of the line eg the central frequency 70 or the central wavenumber v0 strength of the line or intensity S and shape factor or line profile f of the line 0 Each line has a nite width referred to as natural broadening of a spectral line 0 In the atmosphere several processes may result in an additional broadening of a spectral line of the molecules 1 collisions between molecules referred to as the pressure broadening 2 due to the differences in the molecule thermal velocities referred to as the Doppler broadening and 3 the combination of the above processes Lorentz pro le of a spectral line is used to characterize the pressure broadening and is defined as 057 V V02ot2 fLVV0 66 where f V7 v0 is the shape factor of a spectral line v0 is the wavenumber of a central position of a line a is the halfwidth of a line at the half maximum in cm39l often called the line width 0 The half width of the Lorentz line shape is a function of pressure P and temperature T and can be expressed as aPT a0 0 where 10 is the reference halfwidth for STP T0 273K P1013 mb 10 is in the range from about 001 to 01 cm391 for most atmospheric radiatively active gases For most gases n12 NOTE The above dependence on pressure is very important because atmospheric pressure varies by an order of 3 from the surface to about 40 km 0 The Lorentz pro le is fundamental in the radiative transfer in the lower atmosphere where the pressure is high The collisions between like molecules self broadening produces the large line widths than do collisions between unlike molecules foreign broadening Because radiatively active gases have low concentrations the foreign broadening often dominates in infrared radiative transfer Doppler pro le is defined in the absence of collision effects ie pressure broadening as l V V0 2 fDv v0 aDJeXp aD 67 OLD is the Doppler line width 05D akym C where c is the speed of light k1 is the Boltzmann s constant In is the mass of the molecule The Doppler halfwidth at the half maximum is 05D an12 NOTE The Doppler effect comes from random molecular motions If the molecule moves with the thermal velocity Vand emits at the frequency 70 it would appear that it N N V emits at the frequency V 2 V0 1 i where c is the speed of light and Vltlt c The Doppler broadening is important at the altitudes from about 20 to 50 km Doppler line shape Lorentz line shape VVQX Figure 67 Comparison of the Doppler and Lorentz pro les for equivalent line strengths and widths NOTE Line wings are more strongly affected by pressure than Doppler broadening Voigt pro le is the combination of the Lorentz and Doppler pro les to characterize broadening under the lowpressure conditions above about 40 km in the atmosphere ie it is required because the collisions pressure broadening and Doppler effect can not be treated as completely independent processes fVmgVV0 J fLVI V0fDVVIdVI a 1 v v 2 6398 eX dv 013723932 JV39 V0392012 p 01D J no foo NOTE The Voigt pro le requires numerical calculations Lecture 8 Terrestrial infrared radiative processes Part 1Linebyline LBL method for solving IR radiative transfer Obiectives 1 Fundamentals of the thermal IR radiative transfer 2 Linebyline computations of radiative transfer in IR Required reading L02 421423 1 Fundamentals of the thermal IR radiative transfer Recall the equation of radiative transfer Eq 325 a b Lecture 3 for upward and downward intensities in the planeparallel atmosphere 611 T T T 5 Mum 325a d MUT M Jim mo 325b and its solutions Eq326ab Lecture 3 Zquot T 12r lin exp 1 r T T T I I 326 jexplt J2T dr 1 i i T urn 42 110 6Xp Z T T 326b 1 T I I Zjexplt gtJiltr gtdr 0 Infrared radiative transfer in the absorbingemitting atmosphere For a non scattering medium in the local thermodynamical equilibrium the source function is given by the Plank s function B 1T see Lecture 4 and e1 2 lm Zktp where and are the volume extinction and absorption coefficients and k1 is the mass absorption coefficient Assuming that the thermal infrared radiation from the earth s atmosphere is independent on the azimuthal angle p the equation of infrared radiative transfer in the wavenumber domain for the monochromatic upward and downward intensities can be expressed as allT r 13ltwr 3m i V W Iir BAT d r and the solutions as r r IJrIJrmexp r L39 Lifexp 7 BVTr dr Inn m 130 yexp i u T T39 BVTr dr u iiexm 81a 81b 82a 82b To solve Eqs82a b for the entire atmosphere with total optical depth 17Vquot two boundary conditions are required Surface often it can be assumed to be a blackbody with the surface temperature Ts Otherwise the spectral surface emissivity 8V is introduced If rim BAR 11041 Bye Top of the atmosphere TOA 395 0 no downward thermal emission Ito m 0 Using the above boundary conditions the solutions Eqs 2ab for monochromatic upward and downward intensities are 1JrBVrexp T 1 I I 83a ijexp T T Bvr dr lid u 1 J exp T 7 Bvr dr39 83b ll 0 ll The solutions for monochromatic upward and downward intensities can be also expressed in terms of monochromatic transmittance By de nition the monochromatic transmittance is TV TAMI eXp u and the differential form is derw iexpi dr 1 Thus the formal solutions for monochromatic upward and downward intensities given by Eq83ab in terms of transmittance are Mm vamw rm 39 84a I BVTrdTVTdT39rdT39 r 7 JV 1 1 I Ijr yj 3406117 84b 0 NOTE Eq84 a b can be also written in terms of the weighing function which is varvy W 85 Let s rewrite the solutions of the radiative transfer equation for upward and downward de ned as radiances in the altitude coordinate z rv kapgasdz 86 Thus transmission between 2 and 2 along the path at u is 39 1 Z N TVZZIueXp ZJ kvpgmdz 87 and I k z dT Z f i Vpg expkij kvpgasdz 88 dz u 2 Thus T T 1 1 IV w IV mmexp ijva dz39 0 89a 1 z 1 z IeXp I kvpgmdz BVTz39kvpgmdz39 0 x1 z t 1 w 1 z u r I IV L407 j exp jjkvpgmdz VTz kvpgmdz 89b z z39 2 Line bv line LBL quot of radiative transfer in IR LBL method is considered to be an exact computation of radiative transfer in the gaseous absorbingemitting inhomogeneous atmosphere accounting for all known gas absorption lines in the wavenumber range from 0 to about 23000 cm39l I IA39 Strategv to perform LBL ie to solve 17quot 9 9 M for the plane parallel atmosphere For a given wavenumber v For the j th atmospheric layerhomogeneous temperature Tj pressure pj length AZj For n th gas A Absorptlon coefficlent k 1 n 1s L L CIJquot Z kvjnl 2 Sm fvnl 3 1 1 where l l L in the number of absorbing lines of nth gas at a selected v Sm and fml are the intensity and pro le ofthe lth line Optical depth 1 of nth gas of jth layer rk mm mmjumj where MM is the slant path for ngas in jth layer ie the amount of nth gas in jth layer Repeating above calculations for all gases n1 N we find optical depth of j th layer N rm Z 7W n1 Repeating above calculations for all layers j1 J we find optical depth of each layer Using calculated optical depth of each layer we nd the monochromatic upward and downward intensities from Eq 83ab NOTE Similar strategy is used to solve Eq84ab via monochromatic transmission function For nth gas TVyjy Tmquot eXP u and using the multiplication law of transmittance we have the transmittance of j th layer ofN absorbing gases T T T V v1 v2quot T vN o Multiplication law of transmittance states that when several gases absorb the monochromatic transmittance is a product of the monochromatic transmittances of individual gases Tv12N Tv1Tv2TvN 810 For many practical applications one needs to know not monochromatic intensity or ux but intensity or ux averaged over a given wavenumber interval Spectral intensity intensity averaged over a very narrow interval that BV is almost constant but the interval is large enough to consist of several absorption lines Narrow band intensity intensity averaged over a narrow band which includes a lot of lines Broad band intensity intensity averaged over a broad band eg over a whole longwave region We can de ne the spectral transmission function for a band of a width Av as l l l TVu E JTvrdv 3 JeXp rv dv E Jexp kvudv 811 Av Av Av and spectral absorptance is de ned as 1 1 AV 1 7ltugt Ea explt rvgtgtdv Ea explt kvugtgtdv 812 NOTE Spectral intensify requires the calculations of spectral transmission which requires the calculations of monochromatic optical depth which are done with LBL computations LBL spectral resolution Because LBL computes each line of absorbing gases in a nonhomogeneous atmopshere the adequate selection of an integration step ie interval dv is required to calculate the spectral transmittance in the interval Av Av gt dv Because P decreases exponentially with altitude the line halfwidth and hence the integration step should be smaller at higher altitudes in the atmosphere Because of these variable resolutions the absorptions coefficients of two consequent layers must be merged 7 it is done by interpolating the coarserresolution of lower layers into the finerresolution of the higherlevel gt spectral absorptance for a given slant path is computed with the finest spectral resolution gt Continuum Absorption lines may have long wings eg depending on a line halfwidth To simplify calculations the wings of a line are cut at a given distance from the line center Thus the absorption coefficient of the line may be expressed as k S 165 813 where kvc gives the absorption fraction in the wings called continuum absorption CKD Clou Kneiz s and Davies continuum model includes continuum absorption due to water vapor nitrogen oxygen carbon dioxide and ozone The water vapor continuum is based upon a water vapor monomer line shape formalism applied to all spectral regions from the microwave to the shortwave The absorption coefficient for the water vapor continuum is the sum of the self and foreign airbroadening components 0111 k5 Gal17W P PWH 814 Uself where pw and p denote the water vapor partial pressure and the air ambient pressure in atm respectively and cxelf and Fair are the self and airbroadening coef cients for water vapor In the 812 pm region the selfbroadening coefficient is parameterized as Roberts et al 1976 OWN2T al XP3 V 815 where Tr 296 K a418 b5578 and b787x10393 for cmlfin cm2 g 1 atm39l And Smr knelf 0002 at Tr 296 K Moreover cxelf depends on as axelfVT O39e1fVTreXPCTr T 1 where c608 NOTE There are two competing effects on absorption of the water vapor continuum absorption coef cient increases as the temperature decreases but colder atmospheric conditions have less water vapor gt LBL numerical codes LBLRTM and FASTCODE LBLRTM is developed in the ATMOSPHERIC AND ENVIRONMENTAL RESEARCH INC available to the scienti c community httpwwwrtwebaercom General info A radiance algorithm has been used to treat the vertically inhomogeneous atmospheres resulting in substantially improved accuracy and the model is directly applicable to longwave cooling rate calculations A layered atmosphere is used with each layer assumed to be in local thermodynamic equilibrium with respect to absorption in the layer The spectral lines are optimally sampled at each layer using an algorithm that effectively provides optimal sampling over the line An accelerated approximation to the Voigt line Lecture 13 WW precipitation Obiectives 1 Radar basics Main types ofradars 2 Basic antenna parameters 3 Particle backscattering and radar equation 4 Sensing of precipitation and clouds with radars weather radars space radars TRMM and CloudSat Required reading S 81 p401402 57 821 822 823 83 AdditionaVadvanced reading Online tutorial Chapter 3 mm Wwwrrr nrr an r 39 1 ehtml CloudSat web site httpcloudsatatmoscolostateedu TRMM web site httpwwweorcnasdagojpTRMMindeX7ehtm http trmm gsfcnasa gov 1 Radar basics Main types of radars 0 Radar is an active remote sensing system operating at the microwave wavelength 0 Radar is a ranging instrument RAdio Detection And Ranging Basic principles The sensor transmits a microwave radio signal towards a target and detects the backscattered radiation The strength of the backscattered signal is measured to discriminate between di erent targets and the time delay between the transmitted and re ected signals determines the distance or range to the target Two primary advantages of radars all weather and day night imaging Radar modes at aggration o Constant wave CW mode continuous beam of electromagnetic radiation is transmitted and received gt provides information about the path integrated backscattering radiation Pulsed mode transmits short pulses typically 1039610398 s and measures backscattering radiation or echoes as a function of the range Radar range resolution Consider a radar with pulse duration tp dV ttpZ gt R2 c mp2m t gt R ct2 ttp2 gt R1cttp22 Thus radar range resolution is RlRz ctpZ h2 131 where c is the speed of light Problem A police pulsed speedmeasuring radar must be able to resolve the returns from two cars separated by 10 In Find the maximum pulse duration that can be used to prevent overlapping of the returns from the two vehicles Ignore the Doppler effect Solution RlRz 10 m thus tp 210m3108 ms 66710397 s Polarizing Radar has four possible combinations of both transmit and receive polarizations as follows HH for horizontal transmit and horizontal receive VV for vertical transmit and vertical receive HV for horizontal transmit and vertical receive and VH for vertical transmit and horizontal receive Microwave bands used in radar remote sensing see table 33 Ka K and Ku bands very short wavelengths used in early airborne radar systems but uncommon today Xband used extensively on airborne systems for military terrain mapping Cband common on many airborne research systems CCRS Convair580 and NASA AirSAR and spacebome systems including ERSl and 2 and RADARSAT Sband used on board the Russian ALMAZ satellite Lband used onboard American SEASAT and Japanese JERSl satellites and NASA airborne system Pband longest radar wavelengths used on NASA experimental airborne research system T Eyes of radars gt Non imaging galtimeters and scatterometers Altimeters often nadirlooking Operation principle transmit short microwave pulses and measure the round trip time delay to targets to determine their distance from the sensor Applications used on aircraft for altitude determination and on aircraft and satellites for topographic mapping sea surface height measurements from which wind speed can be estimated Example ERS altimeter Figure 13 1 Re ection of an altimeter pulse from a at surface As the pulse advances the illuminated area grows rapidly from a point to a disk as does the retumed power Eventually an annulus is formed and the geometry is such that the annulus area remains constant as the diameter increases The returned signal strength which depends on the re ecting area grows rapidly until the annulus is formed remains constant until the growing annulus reaches the edge of the radar beam where it starts to diIninish to2Hc tpgt tp tp is the duration of the pulse Example TOPEX Poseidon and Jason l radar altimeter sea surface height deviation from the mean in cm Scatterometers Operation principle transmit microwave signal and measures the strength of the backscattering radiation re ection Applications measurements of wind speed and wind direction over the oceans Ground based scatterometers are used extensively to accurately measure the backscatter from various targets in order to characterize different materials and surface types Example NASA Quick Scatterometer QuikSCAT 0 Radar 134 gigahertz llOwatt pulse at l89hertz pulse repetition frequency PRF o Antenna lmeterdiameter rotating dish that produces two spot beams sweeping in a circular pattern QaikSCAT measurement eagabili o 1800kilometer swath during each orbit provides approximately 90percent coverage of Earth s oceans every day 0 Windspeed measurements of 3 to 20 ms with an accuracy of 2 ms direction with an accuracy of 20 degrees Wind vector resolution of 25 km 002 08011999 gt Imaging radars see many examples below dlrnenslon dlmenslonal representataon oflmaglng sensors Sidebaking viewing geametiy afimaging mdar system the alongrtrack dlmenslon parallel to the rght dlrecuon 2 Basic antenna parameters spaee and the uetuahng voltages m the erreurt to whlch rtrs eonneeteol Kari antenna Emumeters m free spaee 1 Frelolpattern 37D quantataes lnvolvlng the vanataon ofEMfleld or EM power as a functlon of the spheneal coordlnates e and p powerpattern 116 p m w srquot and normallzed power pattern 136 zp me go Pme w by apartaele Figure 132 Anlenna power pattern in polar coordinates and in rectangular coordinates Major lobe First null beamwxd h FNRW Halfpower beamwidlh HPBWJ Minor lobes Radiation intensin Half power hcamwidlmlIPBW First null beamwidlMFNBW Majnr lobe Minor lobes Side lobe Back lube A 1r 112 0 112 1r NOTE same name Major lobeMain lobe Main beam Since the difference between the power transmitted by an antenna Pt in W and the power received from backscattering is typically several orders of magnitude the received signal is expressed in Decibels dB Pz39n dB1010g 132 I 2 Antenna gain is de ned as the ratio of the intensity at the peak of the transmission pattern 11 to an isotropic intensity that is derived assuming that the total power Pt in W is distributed equally in all direction I pz 133 Pt 47239R R is the range L V Beam area or beam solid angle in sr is de ned as QA I Pn6pdQ 134 4 The beam area is a solid angle through which all of the power radiated by the antenna would stream if P9 p maintained its maximum value over Q A and was zero elsewhere gt Power radiated in W Pmax9 p Q A The beam area can be approximated by the product of the halfpower beamwidths HPBW see Fig 132 in two principal planes QA z HP HP 135 where 6H is A9 of the HPBW and pH is the Aq of the HPBW 4 Effective aperture A6 in m2 is de ned as 2 AeQA 136 where 7 is the wavelength in m 5 Directivity D Z l dimensionless is de ned as the ratio of the maximum power to its average value D Pmax9 p PMS p Other expressions for the directivity 47239 D Q d1rect1v1ty from pattern 137 A A3 D 47239 7 d1rect1v1ty from aperture 138 gt Friis transmission formula Consider a transmitting antenna of effective aperture Act and receiving antenna with effective aperture Aer The distance between the antennas is R If transmitted power P1 is radiated by an isotropic source the power received per unit area at the receiving antenna if Pt 13 9 47239R2 I and the power available to the receiver is P FAB 1310 But the transmitting antenna has an effective aperture Act and hence a directivity D from Eq138 A D 47239 2122 Thus the power available to the receiver is D times greater 4 7239Aet P FAWD FAB 12 1311 Substituting Eq139 into Eq1311 gives P BABY 472Aet MR2 12 1312 or Pr Aer Aer F R 2 22 1313 t 3 Particle backscattering and radar eguation Recall Lecture 6 in which we introduced introduce the ef ciencies 0r ef ciency factors cross sections and volume coef cients for extinction scattering and absorption Let s introduce backscattering characteristics needed in active remote sensing radar and lidars Differential scattering cross section Cd is de ned as the amount of incident radiation scattered into the direction 9 per unit of solid angle 75 P 1314 0 d 4 where P0 is the scattering phase function Bistatic scattering cross section 6b is de ned as 0391 47F0d 1315 Backscattering cross section 0 is de ned as ab47rad 180 1316 Using Eq1314 Eq1316 can be rewritten as ab UJP 1800 1317 Recall that the incident intensity I and scattered intensity IS by a particle relates as 15 11 1318 where R is the distance from the particle For the backscattering case we can write Fbs 180 Fl 1800 1319 or 11556 180 47rR2 Flob 1320 Thus the physical meaning of the backscattering crosssection is the area that when multiplied by the incident ux gives the total power radiated by an isotropic source such that it radiates the same power in the backward direction as the scatterer For the particle number size distribution Nr the backseattering volume coef cient Kb is kb IabrNrdr 1321 and thus kb kJP 180 1322 where P is the scattering phase function averaged over the size distribution Small size parameter limit Rayleigh limit it can be shown from Mie theory see S 571 that 72395 2 6 ab 7K D 1323 where IK I m z 1 m if the refractive index of the particle andD is the particle m Z 2 diameter gt Radar eguation Consider a transmitting radar with an antenna of effective aperture Act and pulse duration tp or length hctp The radar illuminates an object e g a cloud at the distance R Suppose that the object has the backscattering crosssection called radar crosssection 6r Using the Friis transmission formula we can nd the power intercepted by the object PM as 2 13115 object 13 O39r 1324 Using that the scattering object can be considered as an isotropic source such that it radiates the same power in the backward direction it has directivity D1 and effective aperture Ae 73411 see Eq138 And using the Friis transmission formula we can find the power received by the antenna P Pm by object M i r R212 4 1325 Substituting Eq1324 into Eq1325 we obtain Pr A2 039 FZWF t called the radar equation 1326 where A Aet Aer is the effective aperture of antenna same for transmitting and receiving If the object is a cloud with size distribution Nr and the volume backscattering coefficient kb The power backscattered by the volume IV and received by lidar can be expressed as P A2 Icde F 47ZR4 7 1327 From lidar beam geometry the illuminated volume can be approximated as W m RZGHquHPhZ 1328 and using Eq1321 for kb we have h QHS DHP abrNrdr 1329 Assuming that particle are in the Rayleigh limit and using Eq1323 we have P 7239412 hQHPquP 2 6 K D N D dD 1330 P 416 R2 I l J the above equation can be rewritten as 133 1 where factor C depends on the antenna characteristics and Z J D 6NDdD is called the radar re ectivity factor NOTE Eq1331 is often called the radar equation also We can relate the backscattering coefficient and radar reflectivity as 7139s 2 7139s 2 7139s 2 k 0bDNDdD 7m D6NDdD 71lt J39 D6NDdD 71lt Z 1332 0 If particle are not in the Rayleigh limit andor nonspherical eg ice crystals the effective radar re ectivity factor Ze is introduced 0 In the more general case Eq 1331 must be corrected to account for the attenuation along the path to and from the scattered volume a cloud ie attenuation may arise from absorption by atmospheric gases absorption by cloud drops and precipitation R r r P C Zexp 2 krdr 1333 7 R2 where kg is the extinction coefficient along the path 4 Sensing of precipitation and clouds with radars Principle use a relationship between the radar re ectivity factor Z or Z6 and the rainfall rate Rr mmhour in the form called ZR relationships Z A Rr 1334 where A and b are constants depending on the type of rains Empirical ZR relationships Rr in mmh and Z in mm6m393 Stratiform rain Z ZOORVL6 1335 Orographic rain Z 31 Rl ml 1336 Snow Z 2000Rr2 1337 The power returned to a radar see Eq1331 can be normalized using Eq132 Pin dBZ1010g P 1338 Pref where Prefis the reference power which is often taken to be that power which would be returned if each m3 of the atmosphere contained one drop with D 1 mm Z 1 mm6m393 gt National Weather Service radars httpweather mum 39 39 39 htm The National Weather Service NWS Weather Surveillance Radars WSR are of three types WSR57S WSR74C and WSR88D D stands a Doppler radar Wavelength Dish Diameter cm feet Radar WSR57 103 12 05 or 4 WSR74C 54 8 3 WSRSSD 111 28 157 or 45 Knunan snuey Cheyenne Stating Sleamhnal Springs L Gmm lenn culnmm Springs Gunmsnn sauna Puebla Lamar La Junta Rallarlmale 1mm quotannual Weather Semee KFI G m 15 urc numrznnz Example oflhe WSR radar Image for Colorado for Aprll Z 2002 36 025 mm gt S ace radars TRMM radar and CloudSat radar TRMM radar rst radar m space aurlched m 1997 13 8 GHz 4 3kmfootpr1rlL ZSOm vemcal resolutlorl l 67 HS pulse duratlorl crossLrack scarlrurlg Zl 5km swath Lecture 2 introductory survey 1 Radiation and climate 2 Other important roles of radiation Required reading Keith DW Energetics In Encyclopedia of Climate and Weather Oxford Univ Press pp 278283 1996 Kiehl J T and Trenberth K E Earth39s Annual Global Mean Energy Budget Bull Amer Meteor Soc 78 197208 1997 1 Radiation and climate Radiation is a key factor controlling the Earth s climate Radiation equilibrium at the top of the atmosphere TOA represents the fundamental mode of the climate system Planetary radiative equilibrium over the entire planet and long time interval TOA incoming radiation E TOA outgoing radiation TOA incoming radiation incoming solar radiation 7 re ected solar radiation TOA outgoing radiation outgoing infrared radiation emitted by the atmosphere and surface What is T 0A incoming solar radiation If F0 is the solar constant ie solar ux at the top of the atmosphere then incoming solar radiation per unit area of Earth s surface is F0 Rf 41Re2 F0 4 where Re is the radius of the Earth Assuming F0 1368 Wmz incoming solar radiation is 1368 Wm24 342 W m2 What is re ected solar radiation re ected solar radiation 17 F0 4 where 17 is the global mean planetary albedo Taking 17 03 re ected solar radiation is 1026 W m2 Global mean Earth s energy balance Incomlng 235 0man 342 a Lm ave Bacall11 Ra iarim1 342 w m 235 W Em ed in f 0 I u y tmosphar I Almumhare 165 WIhdew saer by Gwen Atmosphere Sam t 324 330 Back Fanatch 390 68 4 1398 Surface bsozbed by Sums Thermals Evapo Raniailim 324 transpiraan Absumed by Sm m Figure 21 Earth s energy balance diagram from Kiehl and Trenberth 1997 Solar radiation Infrared radiarion re ected by clouds Solar Iadimion 105 to Space km and Emh s surface oimng in by atmosphere Generation kinetic energy 05 23 30 10 0 SDlssipaLion ofkineuc 3 g 39 energy by fricuon 5 m E w Hem released by 3 e g r d v 5 a con ensaucn 0 s u ofwaret venor U 3 3 Infrared radiauun E g 1 8 absorbed by au39nasphere 3 g g E a 14 E 5 39g g e quot5 E E 8 Flux oflatcnt hear 3 quot5 E due 0 evaporaunn Figure 22 Earth s energy balance diagram from Piexoto and Oort 1992 S a al and it oral distribution 0 radiation The planetary albedo is a key climate variable as it combined With the solar insolation determines the radiative energy input to the planet The global annual averaged albedo is approximately 030 The albedo varies quite markedly With geographic region and time of year Oceans have a low albedo snoW a high albedo While the Northern Hemisphere has more land than the Southern Hemisphere the annual average albedo of the tWo hemispheres is nearly the same demonstrating the important in uence of clouds in determining the albedo Figure 23 Examples of monthly mean albedo measured by the NASA Earth Radiation Budget Experiment ERBE satellite Ongoing lunglvave intrareg radiation Low values usually indicate cold tempemtures While high values are Warm areas of the globe The minimum in OLR near the equator is due to the high cloud tops associated With the intertropical convergence zone This minimum migmtes about the equator as seen in the monthly mean maps and is also seen as a maximum in albedo see Figure 23 Notice how it is dif cult to observe the oceanic stmtus regions We observed in the albedo maps This is because the temperature of the clouds is similar to the surrounding oceans making it dif cult to observe Note how the major deserts have their largest OLR during their summer This results from the annual tempemture cycle as the desert surface heats up it emits more longWave mdiation Note also the large emission in the vicinity ofthe oceanic subtroical hi s 30N and 308 JAMUAE I Figure 24 Examples of monthly mean outgoing longWave radiation OLR measured by the NASA Earth Radiation Budget Experiment ERB E satellite Zonal mean mdmmn budget nearly so This 15 not the case when he radAauon gams andlosses are averaged as a or funenon of lamude Regmns between approximately 30m and 30 s gam rad ant energy h rm y Annual Average SUD E EnemVSuvmus E 26 e gt t 22D g Emwssmnu enesma enamv 0 an mspacervamesemsan D enemwussbvmemanet EnEmV E W Dem I e m 9 ncummgsmavenemv39 a an vemesemsanenamvuam 6 VuHhEmane n an en 775 ran 45 e 45 n Lamude Figure 2 5 Senennane representation ofzonal mean radAauon budget

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