In Chapter 3, we derived the Schwarzschild-Milne equations, given the source function, for the net upwelling flux at any atmospheric level arising from flux emitted by the surface and by the atmospheric layers below and above that level. For the troposphere and stratosphere, the bulk of the thermally emitting

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Wavenumber (cm1)

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Wavenumber (cm1)

FlG. 4.17. N2O 7.8 ¡m band rotational line intensity, including contributions from isotopic bands.

atmosphere, collisions determine the populations of the vibrational-rotational states and hence the atmosphere can be taken to be in local thermodynamic equilibrium (LTE). Thus, the source function can be set equal to the Planck function, S\ = B\, which holds below 70 km for most molecules in the Earth's atmosphere (for non-LTE see Lopez-Puertas and Taylor 2001). Equation (3.125) can be simplified by using the diffusivity approximation, discussed in §3.5.5, and the recurrence relation between the exponential integrals E2 and E3.

4.8.1 Upwelling fluxes

We can rewrite, to a very good approximation, eqn (3.125) for the upwelling component at optical depth tx to obtain

C TgX , f+(rx) = n£g\B\(Tg + nBx(T (rA))e-lT—^ drx/^c

where ¡ic =0.6 and Tgx is the total optical depth of the atmosphere at wavelength A, from the top of the atmosphere (TOA) to the surface, and T is the atmospheric temperature. We have also introduced the possibility that the surface does not absorb/emit as an ideal blackbody, at temperature Tg, through the emissivity egx, which is unity for an ideal blackbody and takes values ranging between 0 and 1, this effectively means that the surface can reflect infra-red radiation, with thermal infra-red reflectivity 1-egx.

Furthermore, in order to perform the integrals we need to divide the atmosphere into sufficiently optically thin layers, drx, of homogeneous temperature. We see from the above equation that the integral is essentially a summation of contributions of emission nBx(T(rx))drx/^c from the surface of each layer of homogeneous temperature T, since as we saw in eqn (3.84), the emissivity of an optically thin layer of homogeneous temperature is given by e = dTx/^c. If we now define the transmission between levels rx and tx by t(T\,T'x ) = e-T'x-Txl/^, (4.110)

then we can write for the net upwelling flux fTsX , , fx(Tx) = negxBx(Tg)t(rgx,rx) - nBx[T(rx)jdi(rx,tx). (4.111)

This formulation provides a simpler and smoother way to evaluate the integral in terms of the transmission rather than the optical depth. We note that dt > 0 for atmospheric levels above tx, and vice versa.

4.8.2 Downwelling fluxes

The downwelling flux at the surface is given by iTsX , , f-(Tgx )=/ nBx [T (Tx)]dt(Tx ,Tgx), (4.112)

while the net upward flux at the surface can be written as h(Tg\) = n£g\B\{Tg) - £gXfx (tsx),

noting that only the fraction egX of the downwelling flux is absorbed by the surface. The reflected fraction does not contribute to the net upward flux emitted by the surface.

4.8.3 Outgoing flux at TO A

We can now write for the outgoing flux at the top of the atmosphere

The first term represents the contribution from the surface emission with emis-sivity £g, the second represents that from surface reflection of downwelling flux and the third is that from the atmospheric emission directly to space.

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