## N

where the total columnar number of particles (particles per unit cross-section of atmospheric column or column number density) of all sizes is N = ni. We can also define an effective aerosol radius for the whole aerosol column based on the columnar volume size distribution reff f V (ln r)r3 dlnr f V(lnr)r2dlnr

The effective radius for the fine mode shown in Fig.6.14 is about 0.13 pm throughout the year, while that of the coarse mode varies within the range 2.00-2.32 pm.

Fine mode |
Coarse mode |
Fine + coarse | |||

reff |
VC |
reff |
VC |
VC | |

Winter |
0.131 |
0.025 |
2.004 |
0.057 |
0.082 |

Spring |
0.128 |
0.042 |
2.026 |
0.075 |
0.117 |

Summer |
0.132 |
0.048 |
2.322 |
0.046 |
0.094 |

Autumn |
0.135 |
0.037 |
2.141 |
0.066 |
0.103 |

Annual |
0.132 |
0.038 |
2.123 |
0.061 |
0.099 |

It should be noted that the reff of the coarse mode is slightly larger in summer (the dry season) than in the other seasons due to the absence of wet removal processes of dust. The columnar volume of aerosols, Vc, ranges from 0.025 to 0.048 for the fine mode and from 0.046 to 0.075 for the coarse mode. The maximum columnar volume of fine mode aerosols takes place in summer, owing to the significant contribution of anthropogenic and natural fine pollution particles. The coarse mode has higher columnar volumes in spring and in autumn, when there is strong transport of dust. In spring and autumn, when the frequency of occurrence of dust episodes is highest, the Vc of the coarse mode is larger than that of the fine mode by up to 1.8 times. The very large coarse-to-fine ratio of columnar volume in winter may be attributed to strong processing of aerosols by winter clouds and also to the presence of mineral dust and maritime particles.

### 6.6 Surface reflection

In the calculation of the surface reflectivity we can consider four types of surface; land, ocean, snow, and ice (frozen ocean). The surface albedo can be computed from:

where / is the fraction of the Earth's surface covered by each type of reflecting surface.

### 6.6.1 Snell and Fresnel laws

The ocean reflectivity, Ro, can be computed using Fresnel reflection of natural (unpolarized) light, corrected for a nonsmooth surface for an incidence angle i = cos-1 j from

tan a tan b

with

and r the angle of refraction, as shown in Fig. 6.16. According to Snell's law r is related to the angle of incidence via sin r nsea nair

where nsea = 1-34 is the refractive index of sea water (ratio of speed of light in sea water to speed of light in vacuum), and nair is that of air, given at STP (0 °C, 1 atm) by nair = 1+ A(1 + B/X2), (6.92)

where A and B are given in Table 6.2, and so na wavelength of the sodium D lines).

1.GGG291S at G.59 pm (the

For very small i, that is essentially normally incident direct solar radiation onto a perfectly smooth ocean surface,

and on using Snell's law for small i and r we have i/r « 1-34 we obtain a reflectivity of 0.021. For an ocean we can correct the Fresnel reflectivity to take into account surface roughness

which gives a higher reflectivity of about 0.04 for incident radiation normal to the surface (Kontratyev 1973). The correction factor goes to zero when the radiation is parallel to the ocean surface to maintain a Fresnel reflectivity of unity in this case. The radiation is taken to be isotropically reflected from the non-smooth

incident |
reflected | |

y |
ail | |

sea water | ||

r \ | ||

V transmitted |

FlG. 6.16. The incident radiation at angle i to the sea surface is reflected and transmitted by the sea water. The angle of transmission is the angle of refraction r.

ir r

FlG. 6.16. The incident radiation at angle i to the sea surface is reflected and transmitted by the sea water. The angle of transmission is the angle of refraction r.

ocean surface. There are measurements that indicate that the surface albedo of a rough water surface may not approach the Fresnel limit at low solar elevation angles but can be nearer to 0.3 (Henderson-Sellers and Wilson 1983).

For incident diffuse solar radiation, a mean angle of incidence corresponding to H = 3/5 can be used, which gives an ocean reflectivity for diffuse radiation of 0.055, consistent with the range 0.03-0.10 given for an ocean by Sellers (1965) for cloudy-sky conditions.

Furthermore, we need to include the condition that if the Fresnel reflectivity is greater than the ice or snow reflectivity, which occurs at low solar elevations, then the reflectivities of both snow and ice are set equal to the Fresnel reflectivity. Kuhn (1989) gives observations that clearly show this behaviour for snow cover in Antarctica.

Beyond the dependence of surface reflection on the angle of incidence of the solar radiation, the reflectivity of natural surfaces can be strongly dependent on wavelength, with large differences as we go from the visible to the near-infra-red. Fresh snow has a peak reflectivity of about 0.80 near 0.6 ¡m falling to below 0.10 by 2.0 ¡m. Old snow follows a similar decrease but the reflectivity is generally lower, about 0.7 at 0.6 ¡m. Sand and loam have a reflectivity of about 0.2 near 0.6 ¡m rising to 0.4 near 2.0 ¡m. Vegetation has a reflectivity of about 0.1 below 0.7 ¡m rising rapidly to 0.4 near 0.8 ¡m (see Fig. 10.8 in §10.6).

6.7 Multiple scattering solution for inhomogeneous layers

For a multiple anisotropic scattering solution of the transfer equation, a set of monochromatic radiative flux equations can be solved rapidly for an absorbing-scattering inhomogeneous atmosphere using the delta-Eddington method (Joseph et al. 1976). The delta-Eddington method is an improvement of the Eddington method (Shettle and Weinmann 1970). The latter workers introduced a simplified approach for inclusion of anisotropic scattering in the Eddington approximation using a truncated-phase function, and gave analytical solutions for the two-boundary valued problem for homogeneous atmospheric layers, that is for w and g constant within each layer.

Here we shall derive the radiation diffusion equation for anisotropic scattering, following their truncated-phase function approach, but for nonhomogeneous layers. This allows the inclusion of molecular, aerosol and cloud scattering to be more easily accomplished in each layer. We then give the appropriate simple transformations of the standard radiation transfer parameters g (asymmetry factor), t (extinction optical depth), and w (single scattering albedo) for the delta-Eddington improvement to the solution. The radiation diffusion equation is a second-order ordinary differential equation that can be numerically solved rapidly via the Thomas algorithm. We first re-examine its form for isotropic scattering based on the Eddington approximation discussed in §3.5.6.

### 6.7.1 Isotropic scattering solution

For coherent and isotropic scattering we saw in Chapter 3 that the Eddington approximation results in a second-order differential equation for the mean radiance Jx in the form of a diffusion equation

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