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Understanding Azimuthal Dependence in Scattering Theory and Radiative Transfer

This lesson explores the azimuthal dependence in slab geometry, focusing on the relationships between flux, mean intensity, and the source function. We introduce the asymmetry factor 'g' to characterize angular scattering, along with Legendre polynomials as an orthogonal basis. Key models such as Rayleigh and Mie-Debye phase functions illustrate scattering behaviors. Additionally, we cover the d-Isotropic and d-Two-Term approximations to refine the phase function description. This comprehensive approach is essential for understanding radiation transfer in various mediums.

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Understanding Azimuthal Dependence in Scattering Theory and Radiative Transfer

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  1. METO 621 Lesson 11

  2. Azimuthal Dependence • In slab geometry, the flux and mean intensity depend only on t and m. If we need to solve for the intensity or source function, then we need to solve for t, m and f. However it is possible to reduce the latter problem to two variables by introducing a mathematical transformation.

  3. Azimuthal Dependence • The first moment of the phase function is commonly denoted by the symbol g=c1. • 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 scattering

  4. Legendre 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

  5. Azimuthal Dependence • We can now expand the phase function • Inverting the order of summation we get

  6. 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. • We can now write a radiative transfer equation for each component

  7. Azimuthal Dependence

  8. 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:

  9. Rayleigh Phase Function • The azimuthally averaged phase function is • In terms of Legendre polynomials

  10. Rayleigh Phase Function • The assymetry factor is therefore

  11. Mie-Debye Phase Function

  12. Mie-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 transformations

  13. Scaling Transformations

  14. Scaling 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.

  15. Scaling Transformations • Then the remainder of the phase function is expanded in Legendre Polynomials. • This is known as the d-N approximation • There are simpler approximations

  16. The d-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 • When this phase function is substituted into the azimuthally averaged radiative transfer equation we get:

  17. The d-Isotropic Approximation

  18. The d-Isotropic Approximation

  19. The d-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:

  20. The d-Two-Term Approximation • Substituting into the azimuthally averaged radiative transfer equation:

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