METO 621Azimuthal DependenceSlide 3Legendre PolynomialsSlide 5Slide 6Slide 7Examples of Phase FunctionsRayleigh Phase FunctionSlide 10Mie-Debye Phase FunctionSlide 12Scaling TransformationsSlide 14Slide 15The d-Isotropic ApproximationSlide 17Slide 18The d-Two-Term ApproximationSlide 20METO 621Lesson 11Azimuthal Dependence•In slab geometry, the flux and mean intensity depend only on and  . If we need to solve for the intensity or source function, then we need to solve for ,  and . However it is possible to reduce the latter problem to two variables by introducing a mathematical transformation.p(,)  (2l 1)()Pll02N  1(cos)where Plis the lth Legendre PolynomialAzimuthal Dependence•The first moment of the phase function is commonly denoted by the symbol g=•This represents the degree of assymetry of the angular scattering and is called the assymetry factor. Special values of g are•When g=0 - isotropic scattering•When g=-1 - complete backscattering•When g=+1- complete forward scatteringLegendre Polynomials•The Legendre polynomials comprise a natural basis set of orthogonal polynomials over the domain •The first five Legendre polynomials are•P0(u)=1 P1(u)=u P2(u)=1/2.(3u2-1)•P3(u)=1/2.(5u3-3u) P4(u)=1/8.(35u4-30u2+3)•Legendre polynomials are orthogonal(0  180)klklforluPuduPlklkklfor 0but , 1 where 121)()(2111Azimuthal Dependence•We can now expand the phase function lmmmlllNllmuuuPuPluupp1/120)'(cos)()'(2)()'(x )12(),;','()(cos• Inverting the order of summation we get)()'()12()2(),'( where)'(cos),'(),;','(120120uuluupmuupuupmlmlNmllmmNmmAzimuthal Dependence•This expansion of the phase function is essentially a Fourier cosine series, and hence we should be able to expand the intensity in a similar fashion.I(,u,)  Imm02N  1(,u)cosm(0)• We can now write a radiative transfer equation for each componentAzimuthal Dependence),,()2(4),( where1)-,2N0,1,2,....=(m )()1(),()',',(),;''.(,''4),,(),,(0000/011200upFauXBaeuXuIuupdudauIdudIummSmmmmmmmExamples of Phase Functions•Rayleigh Phase Function. If we assume that the molecule is isotropic, and the incident radiation is unpolarized then the normalised phase function is:pRAY(cos) 34(1 cos2) )"cos()1()"1("2)"(cos)1)("1("143),;","(2/122/1222222uuuuuuuuuupRAYRayleigh Phase Function•The azimuthally averaged phase function ispRAY(u',u) 12d' pRAY02(u',';u,)341 u'2u212(1 u'2)(1 u2)• In terms of Legendre polynomials pRAY(u',u) 112P2(u)P2(u')Rayleigh Phase Function•The assymetry factor is therefore111110),'(''21)'(),'('21uupuduuPuupdugRAYRAYlMie-Debye Phase FunctionMie-Debye Phase Function•Scattering of solar radiation by large particles is characterized by forward scattering with a diffraction peak in the forward direction•Mie-Debye theory - mathematical formulation is complete. Numerical implementation is challenging•Scaling transformationsScaling TransformationsScaling Transformations•The examples shown of the phase function versus the scattering angle all show a strong forward peak. If we were to plot the phase function versus the cosine of the scattering angle - the unit actually used in radiative transfer- then the forward peak becomes more pronounced.•Approaches a delta function•Can treat the forward peak as an unscattered beam, and add it to the solar flux term.Scaling Transformations•Then the remainder of the phase function is expanded in Legendre Polynomials.)(cosˆ)12()1()cos1(2),:','(ˆ)(cosˆ120lNllNNPlffuupp• This is known as the approximation• There are simpler approximationsThe Isotropic Approximation•The crudest form is to assume that, outside of the forward peak, the remainder of the phase function is a constant, Basically this assumes isotropic scattering outside of the peak. The azimuthally averaged phase function becomes)1()'(2),'(ˆfuufuupISO• When this phase function is substituted into the azimuthally averaged radiative transfer equation we get:The Isotropic Approximation1111)',('2)1(),(),()',(),'('2),(),(uIdufauafIuIuIuupduauIdudIuafafadafduIduauIdudIu1)1(ˆ and )1(ˆwhere)',ˆ('2ˆ),ˆ(ˆ),ˆ(or11The Isotropic ApproximationfuupudufISO),'(''21iprelationsh by thegiven ispeak scattering forward theofstrength the111The -Two-Term Approximation•A better approximation results by representing the remainder of the phase function by two terms (setting N=1 in the full expansion). We now get:)()'()12()1()'(2),'(ˆ10uPuPlfuufuuplllllTTAThe -Two-Term Approximation•Substituting into the azimuthally averaged radiative transfer equation:21110 and