Optical thickness and the Schwarzschild equations

Although the radiation field varies in space only as a function of pressure, p, its intensity depends also on direction. Let I(p, n, v) be the flux density of electromagnetic radiation propagating in direction n, measured at point p. This density is just like the Planck function B(v, T), except that we allow it to depend on direction and position. The technical term for this flux density is spectral irradiance. Now we suppose that the radiation propagates through a thin layer of atmosphere of thickness Jp as measured by pressure. The absorption of energy at frequency v is proportional to the number of molecules of absorber encountered; assuming the mixing ratio of the absorber to be constant within the layer for small Jp, the number of molecules encountered will be proportional to Jp, in accord with the hydrostatic law. By Kirchoff's law, the absorptivity and emissivity of the layer are the same; we'll call the value Jtv, and keep in mind that in general it will be a function of v. Let 0 be the angle between the direction of propagation n and the vertical, as shown in Figure 4.1. Now, let At* be the emissivity (and absorptivity) of the layer for radiation propagating in the direction 0 = 0. We may define the proportionality between emissivity and pressure through the relation St* = —Kap/g where g is the acceleration of gravity and k is an absorption coefficient. It has units of area per unit mass, and can be thought of as an absorption cross-section per unit mass - in essence, the area taken out of the incident beam by the absorbers contained in a unit mass of atmosphere. In general k is a function of frequency, pressure, temperature and the mixing ratios of the various greenhouse gases in the atmosphere. Passing to the limit of small Sp, we can define an optical thickness coordinate through the differential equation

dp g

Since pressure decreases with altitude, t* increases with altitude. Radiation propagating at an angle 0 relative to the vertical acts just like vertically propagating radiation, except that the thickness of each layer through which the beam propagates, and hence the number of absorbing molecules encountered, is increased by a factor of 1/ cos 0. Hence, the optical thickness for radiation propagating with angle 0 is simply tv = t*/ cos 0. The equations of radiative transfer can be simplified by using either tv or t* in place of pressure as the vertical coordinate.

The specific absorption cross section k depends on the number of molecules of each greenhouse gas encountered by the beam and the absorption properties characteristic to each kind of greenhouse gas molecule. Letting qi be the mass-specific concentration of greenhouse gas i, we may write n k(v,p, T) = ^ Ki(v,p,T)qi(p) (4.2)

The specific concentrations qi depend on p because we are using pressure as the vertical coordinate, and the concentration of the gas may vary with height;a well-mixed greenhouse gas would have constant qi. The dependence of the coefficients Ki on p and T arises from certain aspects of the physics of molecular absorption, to be discussed in Section 4.4.

Eq. 4.1 defines the optical thickness T*(pi,p2) for the layer between pressures pi and p2. Unless k is constant, it is not proportional to |pi — p2|, but it is a general consequence of the definition that T*(pi,p2) = T*(pi,p')+T*(p',p2) ifp' is betweenpi andp2. Consider an atmosphere with a single greenhouse gas having concentration q(p). Then, if kg is independent of p the optical thickness can be expressed as T*(pi,p2) = kg£ where I is the path, defined by

C P2

The boundaries of the layer are generally chosen so as to make the path and optical thickness positive. The path is the mass of greenhouse gas in the layer, per square meter of the planet's surface, and in mks units has units of kg/m2. If the greenhouse gas is well mixed then I = q • (pi — p2)/g. Now, it often happens that kg increases linearly with pressure - a phenomenon known as pressure broadening (alternatively collisional broadening for reasons that will eventually become clear. If we write kg(p) = KG(po) • (p/po), then we can define an equivalent path

Jp, po g such that t*(pi,p2) = KG(po)£e much as before. The equivalent path still has units of mass per unit area, but because of the pressure weighting will differ from the actual path. For example, if the greenhouse gas is well-mixed then

0 11 / 2 ^ pi — p2 pi + p2 (Ars ie = ~gq 2po(pi — p2) = q—--2p7" (45)

Figure 4.1: Sketch of the radiative energy balance for a slab of atmosphere illuminated by incident radiation from below.

Figure 4.1: Sketch of the radiative energy balance for a slab of atmosphere illuminated by incident radiation from below.

The equivalent path is thus the actual path weighted by the ratio of the mean pressure to the reference pressure.

Consider now the situation illustrated in Fig. 4.1, in which radiation at a given frequency and angle is incident on slab of atmosphere from below. In general, part of the incident radiation is scattered into other directions. However, for infrared and longer wave radiation interacting with gases, such scattering is negligible; scattering is also negligible for infrared interacting with condensed cloud particles made of substances such as water, which are strong absorbers. Here, we shall neglect scattering, though it will be brought back into the picture in Chapter 5. The radiation at the same angle which comes out the top of the slab is then the incident flux minus the small amount absorbed in the slab, plus the small amount emitted. Thus

I (r* + ¿r*,n,v) = or, passing to the limit of small Jr*,

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