## Di 2na1

d3 = -^26(1 + RsX) - RsX(6.139) while for the upper boundary condition di = -2a, (6.140)

6.7.3.2 Enhancement factor for photolysis The enhancement factor for photolysis is given by

loSQX

where h0SqX is equal to the incoming radiation at the top of the atmosphere (TOA). At TOA the boundary condition gives JX = -2HX so that the upwards diffusive flux is then equal to 2nJX. The maximum value for the enhancement factor for photolysis at TOA is then obtained for a purely scattering atmosphere that overlays a surface with reflectance RsX = 1. Under these conditions the upwards diffusive flux is equal to the incoming radiation so that 2nJX = h0SqX and so the maximum value of f at TOA is 3.

Under the above scattering conditions, JX becomes constant deep in the atmosphere for large scattering optical depths, where the direct component vanishes, and is given by

so that the maximum value for f is then attained for |o = 1 and is equal to 5. We note that the ratio of Jtoax to JdeepX is then

Jtoax 2

and hence, when ¡i0 = 1 the above ratio is | (cf. the limb-darkening law we discussed in §3.5.10).

6.7.3.3 Delta-Eddington approximation Joseph et al. (1976) improved the truncated phase function approach by including a strong forward scattering component in the phase function p(ti) = 2f(1 - costf) + (1 - f)(1 + 3gcosti), (6.146)

where f is the fraction of scattered photons in the forward direction. Assuming that the original phase function that they were approximating was the Henyey-Greenstein phase function, they obtained f= g2. They then found simple transformations that significantly improved the truncated-phase function approach. These were

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