In Fig. 6.3 is shown the solar irradiance, at low resolution (see §5.2.4), and the atmospheric molecules that absorb it, mainly in the near-infra-red. In Table 6.1 are shown the main absorbing molecules in each spectral interval, including Rayleigh scattering, from the ultraviolet to the near-infra-red.

The most important molecular absorption relevant to the 0.20-0.85 ¡m range of solar radiation is that of O3. The absorption is temperature-dependent in the range 0.20-0.35 ¡m (Hartley-Huggins bands), corresponding to about 4.5% of the incoming flux. The ozone layer also absorbs in the visible between 0.45 and 0.85 ¡m (Chappuis bands), corresponding to about 55.4% of the incoming solar

0.25

0.20

0.15

0.10

0.05

Wavelength (urn)

FlG. 6.3. Low-resolution solar irradiance and atmospheric absorption. (Source: CSR University of Texas, based on the original from Valley 1965, Air Force Cambridge Research Laboratories)

0.20

0.15

0.10

0.05

Wavelength (urn)

Interval |
Wavelength (^m) |
% Solar flux |
Molecules/scattering |

1 |
0.10-0.20 |
0.00644 |
NH3, CH4, C02, H20, 02, Rayleigh |

2 |
0.20-0.35 |
4.46 |
O3, Rayleigh |

3 |
0.35-0.45 |
10.63 |
Rayleigh |

4 |
0.45-0.85 |
44.77 |
O3, H2O, Rayleigh |

5 |
0.85-1.00 |
9.48 |
H2O, Rayleigh |

6 |
1.00-1.04 |
2.12 |
H2O |

7 |
1.04-1.22 |
7.55 |
H2O |

8 |
1.22-1.58 |
9.19 |
H2O, CO2 |

9 |
1.58-1.95 |
4.81 |
H2O |

10 |
1.95-2.12 |
1.24 |
H2O, CO2 |

11 |
2.12-2.61 |
2.41 |
H2O |

12 |
2.61-2.92 |
0.92 |
H2O, CO2 |

13 |
2.92-3.88 |
1.34 |
H2O, CH4, NH3 |

14 |
3.38-4.60 |
0.40 |
CO2 |

15 |
4.60-9.50 |
0.64 |
H2O, CH4, NH3 |

radiation. Absorption cross-section data in the Chappuis bands (0.40-0.85 pm) are shown in Chapter 7. Detailed ultraviolet-visible cross-section data are available from the Jet Propulsion Laboratory (JPL 2006) and there is also available the MPI-Mainz-UV-VIS Spectral Atlas that is a comprehensive collection of absorption cross-sections for 630 molecular species. Both databases can be ac cessed via the web. More details are given in Chapter 7 in relation to atmospheric photochemistry.

In the UV-visible spectral range, the ozone cross-sections are sufficiently smooth to allow the wavelength range to be finely subdivided (e.g. 200 spectral intervals) into monochromatic absorption optical depths, from Lyman-a at 0.1216 ¡m out to 0.85 ¡m, not including the highly structured Schumann-Runge (S-R) bands (0.18-0.20 ¡m) of O2, which require special treatment. More details are given regarding the S-R bands in relation to the photolysis of molecules in Chapter 7. Having the fine-resolution spectral distribution of ozone UV-visible absorption cross-sections one can incorporate these in a multiple scattering and absorption code (§6.7) to include the effects of ozone on the transfer of solar radiation through the atmosphere.

For ozone in relation to the Earth's radiation budget, one can make a quick estimate of the flux-mean atmospheric transmission, from the expressions given below. The transmitted fractions of the total incoming solar radiation due to the absorption of visible and UV radiation by ozone are given, respectively, by where Wo3 represents the total atmospheric ozone amount in atm-cm STP. The minimum value of t'UV is thus 0.955, while for t'vis it is 0.446.

The transmission, t, for each near-infra-red interval (Table 6.1) where rotational lines of molecular vibrational-rotational bands become important, can be computed using the HITRAN database by subdividing the solar near-infra-red spectral intervals into higher-resolution subintervals and then computing the transmission within each subinterval as a function of pressure, temperature and absorber amount for each molecule (§4.7). This approach is now computationally feasible and preferred. However, for climate models (e.g. general circulation models, §11.4) such an approach may not be feasible and a simpler approach is needed. We note that when the optical depth varies non-linearly with absorber amount we cannot simply divide the atmosphere into layers and then multiply their transmissivities. Methods that are employed to overcome this problem are the exponential-sumfit, k-distribution and the correlated-k approximation, according to which the average transmissivity within a spectral interval is obtained from c = 1 - 0.023 (W03/i)018 ,

n for an absorber amount y, in terms of effective monochromatic absorption coefficients, kn, and associated discrete probability distribution weights an, with the normalization

The sets of kn and an for each spectral interval are not unique and can be obtained using, for example, the Fibonacci minimization technique (Vardavas 1989). For inhomogeneous layers, the correlated-fc approximation can be used to allow kn to be temperature and pressure dependent and the an are fixed and correspond usually to Gaussian-Legendre quadrature points.

The monochromatic radiation-transfer equation is then solved for each optical depth, Tn = kny for incoming solar flux anFi, where Fi is the flux contained in each spectral interval i. For overlapping bands of two different molecules, with absorber amounts y and v, the transmission can be taken to be tnm = anpm exp(-kny) exp(-/mv), (6.5)

with the normalization condition

and the transfer equation is solved for anpmFi with optical depth Tnm = kny + /mv for each combination of n and m. This approach can be extended to overlapping bands of more than two types of molecules. One can then use a monochromatic multiple-scattering radiation-transfer code to solve the radiation-transfer equation, as discussed in §6.7. Coefficients and weights for the near-infra-red intervals of Table 6.1 for water vapour, carbon dioxide, methane and ammonia are given in Vardavas and Carver (1984) based on laboratory measurements at 1 atm and 300 K.

6.2.2.1 Water vapour has significant absorption bands in the near-infra-red (0.85-5.0 /m), which corresponds to about 40.1% of the incoming solar radiation. The fraction of the total incoming solar flux absorbed by a H2 O layer can be quickly estimated from t'w = 1 - 0.106 (Who/m)0'31 , (6.7)

where WH2O is the amount of H2O, g cm~2, in the layer. The above simple expression is based on the data of Wyatt et al. (1964).

6.2.2.2 Carbon dioxide absorption of solar radiation takes place throughout the atmosphere. We can neglect the UV absorption of CO2 that occurs below 0.2 «m where the solar flux is less than 0.01% of the incoming flux but we need to include the near-infra-red absorption (Stull et al. 1964). The fraction of the total incoming solar flux transmitted by a CO2 layer can be quickly estimated from t'c = 1 - 0.015 (wco2 /M)0'263 , (6.8)

where WCO2 is the total amount of CO2, g cm-2, in the layer. For a clear sky we have one CO2 layer below the ozone layer, while for a cloudy sky we have two, one above and one below. If po represents the surface pressure at sea level then the amount of CO2 above a cloud layer with cloud-top pressure pcia is WCO2 (pcia/po) while in the layer below the cloud layer the amount is WCO2 (1 — pclb/po), where pclb is the cloud-base pressure.

6.2.2.3 Ammonia absorbs in the near-infra-red intervals 1300-2000 cm-1 (6.14 Mm) and 3100-3500 cm-1 (3.03 ^m). The mean transmission of each band can be obtained from the following simple analytical fits to the laboratory data (France and Williams 1966) in the form t = exp(—wa(w)). (6.9)

For the ammonia 6.14 Mm band a(w) = 38.09w-0'26 with a corrected absorber amount w = w0(pe/po)C with C = 0.6 andpe = pN2 + (B — 1)pnh3 and B = 5.77. For the 3.03 Mm band of ammonia we have a(w) = 32.68w-0178 with C = 0.3 and B = 6.1.

6.2.2.4 Methane absorbs in the intervals 1100-1750 cm-1 (7.66 Mm) and 27003300 cm-1 (3.31 Mm). For the 7.66 Mm methane band we have, from fits to laboratory measurements (Burch and Williams 1962), a(w) = 8.22w-0'46 with C = 0.5 and B = 1.38, while for the 3.31 Mm band a(w) = 9.97w-0'39 with C = 0.45 and B = 1.3.

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