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 cv kB δT , 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 cv kB δT + 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 δT δ ln(∆V ) R = N cv kB +p . (88) δt δt From the ideal gas law, ∆V = ∆N kB T /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 c p kB
∂T . ∂t
(90)
The specific heat at constant pressure is cp = cv + 1 =
f + 1 = 3.5, 2
(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 R RT ∂T = = . ∂t c p kB N cp p
(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 33
