where tox is the total optical depth of the atmosphere and ¡1 > 0 defines the upward non-vertical direction of the emitted radiation. If we perform the transformation w = — dw =--Trd/j, d/j, =--tt

and the flux can be computed from fx = nBx2E3 (tox), (3.86)

where nBx is the surface spectral emission (W m-2jm), integrated over all directions, and the special function En is the exponential integral defined by

which obeys the recurrence relations d E 1

The diffusivity approximation is to define a characteristic direction by jc such that the total flux arriving at TOA in all directions j is equivalent to radiation traversing the atmosphere in one effective direction. This characteristic direction is defined by the approximation

where ^c = 0.6, and hence it is customary to multiply the atmospheric optical depth by 1.66 to account for the transverse transfer of radiation through the atmosphere. As can be seen, when ¡j, approaches zero the effective optical depth approaches infinity, i.e. we have totally tangentially directed radiation that of course is never transmitted to TOA. On the other hand, as j approaches zero, the projection of the emitting surface area approaches zero in any case.

3.5.6 The Eddington and diffusion approximations

Deep (optically) within a static planetary atmosphere the radiation field tends towards isotropy, Ix(j) = Ix, if the scattering is coherent and isotropic. Near the surface directional effects become important. Eddington (1926) allowed for the anisotropy of the radiation field by expanding the radiance Ix (j) in terms of Legendre polynomials Pn(j) so that

Ix (j) = axPo (j) + bxPi(j) + cxP2(j) + ..., (3.88)

where j = cos 0, with the first three Legendre polynomials given by

, and showed that deep within an atmosphere a good approximation is to neglect higher-order terms beyond the second. Thus, the Eddington approximation is to set

For coherent and isotropic scattering, by particles within a static atmosphere, the source function is isotropic and given by

where is the single-scattering albedo. The equation of transfer for a planeparallel atmosphere can then be written as

We now introduce the three moments of the radiation field used by Eddington

1 f1

with the corresponding equations of transfer

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