4.1 Radiative cooling

4.1 Radiative cooling

Consider a small parcel of air of volume ∆V , containing ∆N molecules at a pressure p and temperature T . The heating rate R of (50) will add the heat increment δQ = R∆V δt to the parcel in the time interval δt. According to the first law of thermodynamics, one part of the radiant heat increases the internal energy, ∆U , by an amount δ∆U = ∆N cvkBδT , where cv is the specific heat per molecule at constant volume (in units of kB) and δT is the increase of the parcel temperature. The other part of the radiant heat provides the work, δW = p δ∆V , done on the surrounding air by the expansion, δ∆V of the parcel volume. So the first law of thermodynamics is

R∆V δt = δU + δW = ∆N cvkBδT + p δ∆V . (87)

Dividing both sides of (87) by ∆V δt and recalling that the parcel number density is N = ∆N/∆V , we find

R = N cvkB δT δt + p δ ln(∆V ) δt . (88)

From the ideal gas law, ∆V = ∆N kBT /p, so

δ ln(∆V ) = δ ln(∆N kB/p) + δ ln T = δ ln T . (89)

Here we noted that under the conditions of near hydrostatic equilibrium that we are considering, δ∆N = 0 and δp = 0 since heating (or cooling) a parcel dry air does not change the number of molecules, ∆N , and does not change the weight of the column of air above, which fixes the pressure p. Combining (88) with (89), and letting δT /δt → ∂ T /∂ t we find

R = N cpkB ∂ T ∂ t .

The specific heat at constant pressure is (90)

cp = cv + 1 = f 2 + 1 = 3.5, (91)

where f is the number of degrees of freedom per molecule. In (91) f = 5 is the sum of 3 translational and 2 rotational degrees of freedom of the diatomic molecules, N2 and O2. There is negligible vibrational excitation. From (90) and the ideal gas law (8), we can write the rate of change of the temperature

∂ T ∂ t

R cpkBN RT cpp . (92)

The temperature-change rates (92) associated with the radiant heating rates R of Fig. 14 and Fig. 15 are shown in Fig. 17. Since the heating rate R from thermal radiation is negative over most of the atmosphere, so is the temperature-change rate. The rapid increase of these “diabatic” cooling rates with increasing altitude is mainly due to the rapid decrease of the number density N , illustrated in Fig. 3. The local peak at the stratopause is due to a

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