## Computation of fluxes

4.10.1 Data requirements

To compute the Earth's longwave radiation budget one needs the following atmospheric, Earth surface, and cloud properties:

• Molecular mixing ratio vertical profile

• Molecular band absorption

• Atmospheric vertical temperature and pressure profiles

• Earth's surface temperature and emissivity

• Cloud infrared optical depth

• Cloud-cover fraction

• Cloud-top temperature and pressure

• Cloud-base temperature and pressure in order to compute, at each wavelength, the thermal fluxes at the Earth's surface, within the atmosphere and at TO A. The next step is to integrate over wavelength to obtain the total outgoing longwave radiation (OLR) and the surface radiation budget (SRB). The difference between the net upward flux at the surface and the outgoing flux at TOA is a measure of the atmospheric greenhouse effect under clear-sky conditions. The integration over wavelength means that the radiative properties of the Earth-atmosphere-clouds system at each wavelength need to be known. This is a formidable task and one needs to define the spectral resolution of one's radiation-transfer model based on the available spectral properties of the system. For molecular absorption one can either attempt a line-by-line transfer (use of HITRAN) or use a broad-spectral-interval approach by dividing the Planck function into intervals with significant molecular band absorption, as shown in Table 4.2. This simpler approach involves the computation of the various thermal infra-red fluxes at wavelengths at the midpoints of each spectral interval.

### 4.10.2 Curtis-Godson approximation

If Uj is the wave number at the centre of the interval j and Awj is the width of the interval, then we can use the mean band transmission tik for this interval due to some molecule k. We can then derive a mean optical depth, Tjk for each spectral interval j for each molecule k from

then sum these optical depths for each molecule k to obtain the total optical depth for the interval j, and hence the mean transmission tj. We can evaluate the mean transmission of the atmosphere by dividing it into pressure levels so that each atmospheric layer is sufficiently optically thin for all j. We can then compute the transmission between each pressure level tj (p ,p). Note that this transmission is both temperature and pressure dependent since the molecular band absorptances are so dependent. We thus need to obtain a representative temperature, Tr, and pressure, pr, between the two atmospheric levels we wish to evaluate the transmission. One approach, the Curtis-Godson approximation, is to correct for inhomogeneous atmospheric layers by weighting the pressure or temperature by the absorber amount over the transmission path within the atmosphere so that

W g where pe is the effective broadening pressure and w is the absorber amount (g cm~2) or column density given by w=[ri—, (4.118)

g with n the mixing ratio by mass, i.e. g of molecule per g of air.

We can use the flux equations of §4.8 to compute the various fluxes at each wave number Uj, by replacing with nBj = n JAu. Bu dw and then sum over all spectral intervals j to obtain the total fluxes. In this way, the clear-sky downwelling flux at the Earth's surface for spectral interval j, would be computed, for example, from r Pg

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