Note that we have defined ¡j, = cos 0 such that ¡j, > 0 in the outward direction while the optical depth is zero at TOA increasing towards the surface. The solution for a layer defined by the boundaries zmax to z, as shown in Fig. 3.6, is then ix(tx, ¡) = Ia(0, ^, (3.80)

where j < 0 for incoming radiation with the upper boundary value Ix(0,j) at

3.5.4 Solution for thermal emission

A planetary atmosphere can absorb and emit thermal infrared radiation, given that it has heteronuclear molecules (e.g. H2O, CH4, O3, CO2) that can absorb thermal infra-red radiation. These are molecules that have an electric dipole moment due to residual positive and negative charges on the different types of atoms that form the molecule. For homonuclear molecules (e.g. N2 and O2) there is no dipole moment and hence these molecules do not interact with the thermal radiation field. The equation of transfer when we have both absorption and emission, can be written as

since the source function, Sx is equal to the Planck function (i.e. is isotropic) for thermal radiation at sufficiently high pressures, and hence not valid near TOA. If we consider a single layer with a homogeneous temperature, T, and optical depth tx, then the emission from either the top (or base) of the layer is given by the solution h(rx,v) = Bx(T)(1 - e(3.82)

with ¡1 > 0. We immediately see that if the optical depth is large, the radiance from the top of the layer is h (Tx ) = BX(T), (3.83)

and hence an optically thick (tx ^ 1) atmospheric layer emits like a blackbody. If the layer is optically thin (tx ^ 1) then we can use a Taylor-series expansion of the exponential function, and keep the first two terms to obtain

3.5.5 The diffusivity approximation

Let us consider the case of atmospheric absorption of thermal radiation above a planetary surface taken to emit isotropically as a blackbody at temperature To. The total upward flux arriving at the top of atmosphere, TOA, that was emitted by the surface is fx = 2nBx(To) i e-Tx/^MdM (3.85)

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