■'tv where t'v is a dummy variable and is the optical thickness of the entire atmosphere, i.e. T*(ps, 0)/cos 0. Note that depends on v in general, though we have suppressed the subscript for the sake of readability. The top of the atmosphere (p = 0) is at tto. The physical content of these equations is simple: I+(tv, v) consists of two parts. The first is the portion of the emission from the ground which is transmitted by the atmosphere (the first term in the expression for I+). The second is the radiation emitted by the atmosphere itself, which appears as an exponentially-weighted average (the second term in the expression for I+) of the emission from all layers below tv, with more distant layers given progressively smaller weights. Similarly, I_(tv, v) is an exponentially-weighted average of the emission from all layers above tv , plus the transmission of incident downward flux. The atmospheric emission to space will be most sensitive to temperatures near the top of the atmosphere. This emission will dominate the OLR when the atmosphere is fairly opaque to the radiation emitted from the ground, whereas the transmitted ground emission will dominate when the atmosphere is fairly transparent. The downward radiation into the ground will be most sensitive to temperatures nearest the ground.

In the long run, it will save us some confusion if we introduce special notation for temperatures and fluxes at the boundaries; this will prove especially important when there is occasion to switch back and forth between pressure and optical thickness as a vertical coordinate. The temperature at the top of the atmosphere (p = 0 or t = tto) will be denoted by TTO,and the temperature of the air at the bottom of the atmosphere (p = ps or t = 0) will be called Tsa. For planets with a solid or liquid surface this is the temperature of the gas in immediate contact with the surface. For such planets, one must distinguish the temperature of the air from the temperature of the surface (the "ground") itself, which will be called Tg. The outgoing and incoming fluxes at the top of the atmosphere will be called (v) and Z_,TO(v) respectively, while the upward and downward fluxes at the bottom of the atmosphere will be called 1+,s(v) and /_,s (v)

For planets with a liquid or solid surface, we require that 1+,s(v) be equal to the upward flux emitted by the ground, which is e(v)B(v, Tg), where e(v) is the emissivity of the ground. Continuity of the fluxes is required because, the air being in immediate contact with the ground, there is no medium between the two which could absorb or emit radiation, nor is there any space where radiation "in transit" could temporarily reside. We generally assume that there is no infrared radiation incident on the top of the atmosphere, so that the upper boundary condition is = 0. The incident solar radiation does contain some near-infrared, but this is usually treated separately as part of the shortwave radiation calculation (see Chapter 5). For planets orbiting stars with cool photospheres, such as red giant stars, it might make sense to allow to be nonzero and treat the incoming infrared simultaneously with the internally generated thermal infrared. Since the radiative transfer equations are linear in the intensities, it is a matter of taste whether to treat the incoming stellar infrared in this way, or as part of the calculation dealing with the shorter wave part of the incoming stellar spectrum.

For gas or ice giant planets, which have no distinct solid or liquid surface, we do not usually try to model the whole thermal structure of the planet all the way to its center. It typically suffices to specify the temperature and convective heat flux from the interior at some level which is sufficiently deep that the density has increased to the point that the fluid making up the atmosphere can be essentially considered a blackbody. Once the optically thick regime is reached, one doesn't need to know the temperature deeper down in order to do the radiation calculation, any more than one needs to know the temperature of the Earth's core to do radiation on Earth.

The weighting function appearing in the integrands in Eq. 4.9 is the transmission function. Written as a function of pressure, it is

Tv (p1,p2) is the proportion of incident energy flux at frequency v which is transmitted through a layer of atmosphere extending from p1 to p2; whatever is not transmitted is absorbed in the layer. Note that (p,p')dTV = (with p held constant), if p < p', and (p,p')dTV = —if p > p'. Using this result Eq. 4.9 can be re-written

1_(tv, v) = Z_,ro(v)Tv(0,p) + / nB(v,T(p'))dTv(p,p')

In the integrals above, the differential of Tv is meant to be taken with p held fixed. Integration by parts then yields the following alternate form of the solution to the two-stream equations:

1+(p, v) = nB(v,T(p)) + (/+,s(v) - nB(v,Tso))Tv(p,ps) + / nT„(p,p')dB(v,T(p'))

1-(p, v) = nB(v,T(p)) + (/_,ro(v) - nB(v,Tro))Tv(0,p) - T nTv(p,p')dB(v,T(p'))

Neither of these forms of the solution is particularly convenient for analytic work, but either one can be used to good advantage when carrying out approximate integrations via the trapezoidal rule (see Section 4.4.6). For analytical work, and some kinds of numerical integration, it helps to rewrite the integrand using dB = (dB/dT)(dT/dp')dp'. The result is fPs dB dT

v) = nB(v,T (P)) + (I+,s(v) - nB(v,Tsa))Tv (P,Ps) + nTv (P,P') dT |T(P') dP'

A considerable advantage of any of the forms in Eq. 4.11, 4.12 or 4.13 is that the integration variable p7 is no longer dependent on frequency. This will prove particularly useful when we come to consider real gases, for which the optical thickness has an intricate dependence on frequency. The first two terms in the expression for the fluxes in either Eq. 4.12 or 4.13 give the exact result for an isothermal atmosphere; in each case, the first of the two terms represents the contribution of the local blackbody radiation, whereas the second accounts for the modifying effect of the boundaries. The boundary terms vanish at points far from the boundary, where T is small. Note that the boundary term for I+ vanishes identically if the upward flux at the boundary has the form of blackbody radiation with temperature equal to the surface air temperature. For a planet with a solid or liquid surface, this would be the case if the ground temperature equals the surface air temperature and the ground has unit emissivity.

The main reason for dealing with radiative transfer in the atmosphere is that one needs to know the amount of energy deposited in or withdrawn from a layer of atmosphere by radiation. This is the radiative heating rate (with negative heating representing a cooling). It is obtained by taking the derivative of the net flux, which gives the difference between the energy entering and leaving a thin layer. The heating rate per unit optical thickness, per unit frequency, is thus

This must be integrated over all frequencies to yield the net heating rate. For making inferences about climate, one ordinarily requires the heating rate per unit mass rather than the heating rate per unit optical depth. This is easily obtained using the definition of optical depth, specifically,

H = gd" (I+ -1-) = g^/- (I+ -1-) = "IT Hv (4.15)

dp dp dTv cos 0

When integrated over frequency this heating rate has units W/kg. One can convert into a temperature tendency K/s by dividing this value by the specific heat cp.

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