If we set ts = 0.85, for the total grey optical depth of the present atmosphere, and = 255 K then we obtain To = 214 K, Ts = 288 K and the near-surface air temperature is then T(ts) = 263 K. We see that there is a discontinuity of 25 K

between the surface temperature and the near-surface air temperature. We also see that for ts ^ 1, the atmospheric layer above the blackbody surface is also a blackbody, eqn (3.83), and so from the above equations we have T(ts) « Ts.

Thus, in radiative equilibrium, the temperature rapidly decreases with altitude so that the atmospheric density increases with altitude. We thus have colder more dense layers over warmer less dense layers. This initiates convective transport of heat as the colder layers sink towards the surface and the warmer layers rise up in the atmosphere. If we now define the adiabatic atmospheric lapse rate by r = -dT/dz and the corresponding radiative equilibrium value by rrad, then when rrad > r the atmosphere is unstable against convection and hence its temperature is determined by convective equilibrium as given by the adiabatic lapse rate. We can then use this criterion to determine the altitude when the atmosphere is stable against convection, that is, it is in radiative equilibrium.

The lower atmosphere or troposphere is considered to be in convective equilibrium. The temperature structure is then determined by the surface temperature, Ts, and the atmospheric lapse rate, r. The adiabatic lapse rate in saturated air was derived in Chapter 2 and is given by

with rdry = g/cpd = 9.76 K/km, where g is the gravitational acceleration and cpd is the specific heat of dry air at constant pressure for the gas mixture that constitutes the atmosphere. In the presence of large amounts of water vapour cp will increase above the dry air value. In §2.6, we obtained an expression for the tropospheric lapser rate when water vapour is a minor constituent, as it is in the present atmosphere. For atmospheres at higher surface temperatures, as would be for elevated CO2 levels (possibly in the past during the Precambrian period 4.5-2.5 billion years ago) or larger solar incoming flux (expected in the future on timescales of billions of years as the Sun ages), the atmospheric water-vapour content can increase substantially with the Earth's surface temperature. For the general case, the moist lapse rate factor for a reversible saturated adiabatic expansion is given by (Iribarne and Godson 1981)

The saturated lapse rate is always less than the dry value, and the ratio fc can fall as low as 0.21 in a hot atmosphere of 400 K (Fig. 11.3). The total atmospheric pressure is p = pd + pv, rw = epv/pd is the water-vapour mixing ratio by mass, where e = MH2O/Mair = 0.62, is the ratio of the molecular weight of water vapour (18.02 g mole-1) to that of dry air (28.96 g mole-1). The total water mixing ratio rtw includes liquid water that for pseudo-adiabatic conditions can be set equal to rw. The specific heat of dry air, cpd, is equal to 0.24, R = 1.9865/M and Rd = 1.9865/Mair are the gas constants, for the mixture and dry air, respectively, all in cal g-1K-1. For water vapour, the gas constant is Rv = 0.1103 and the specific heat capacity is cpv = 0.443. The specific heat capacity of water is cw = 1.008, while for ice there is a very weak temperature dependence ci = 0.503 + 1.767 x 10-3(T - To). (11.14)

Lv is the latent heat of evaporation (cal g-1) over a water body (T > To), or that of sublimation over a frozen water body (T < To) and is given by Kirchoff's law dL12 ,,, , rN

where c1 and c2 correspond to phases 1 and 2 of the water substance, and we note that a change of phase occurs at constant pressure. On integrating, we get

Liv = 667.0 - 0.06(T - To) - 8.835 x 10-4(T - To)2 T < To, (11.17)

where To = 273.15 K = 0.0 °C, is the freezing point of fresh water and the latent heat of melting is 79.7 cal g-1 at T = To. We note that sea water freezes at -1.9 °C. Assuming a linear dependence of the latent heat on temperature, we can integrate the Clasius-Clapeyron equation (eqn (2.14)) to obtain the saturation water vapour pressure, at T, in the form lnpv = a1 - a2/T + a3 ln T. (11.18)

To a very good accuracy we can set a1 = 55.49369, a2 = 6814.413, and a3 = -5.12239, for T > To, and a1 = 27.54365, a2 = 6195.730, and a3 = -0.543971, for T < To. At T = To we have pvo = 6.107 mbar. In Fig. 11.3 are shown; the growth of the water vapour pressure with temperature for saturated conditions (relative humidity, rh = ph20/pv, equal to unity), the rise in the atmospheric specific heat capacity, the decrease in the mean molecular weight and the decrease in the moist lapse rate for a dry air atmosphere component at 1013.25 mbar (1 atm). As expected, the water-vapour pressure rises rapidly to 1 atm at the standard boiling point of100 °C (373.15 K), and by 400 K it reaches 2342 mbar, while the moist lapse rate remains fairly constant above 100 °C at 2.1 K km-1. Thus, by 400 K, under saturated conditions, the atmospheric pressure would rise to 3.3 atm. The discontinuity in the moist lapse rate at 273.15 K is due to the phase change from ice to liquid water.

250 300 350

Temperature (K)

250 300 350

Temperature (K)

250 300 350

Temperature (K)

250 300 350

Temperature (K)

S 26

250 300 350

Temperature (K)

250 300 350

Temperature (K)

flg. 11.3. Variation of saturated water-vapour pressure, specific heat capacity, mean molecular weight, and moist lapse rate for a dry atmosphere component at 1013.25 mbar.

The upper atmosphere is in radiative equilibrium. This is achieved by a balance between solar radiation heating, mainly due to atmospheric ozone absorption (see Chapter 6) and radiative cooling to space by carbon-dioxide emission in the optically thin upper atmosphere (see Chapter 4). At low pressures the coupling between the infra-red rotational excited states of a molecule and the thermal energy of the atmosphere is weakened and radiative transitions between discrete energy states are faster than collisional excitation and de-excitation. The populations of the molecular states are then determined primarily by radiation processes rather than collisional ones and the atmosphere is not in local thermo-dynamic equilibrium (Non-LTE). The problem of computing the atmospheric temperature then becomes more complex (see Lopez-Puertas and Taylor 2001). Effects of Non-LTE for the present atmosphere become important above about 70 km altitude. Parameterizations for computing the radiative equilibrium temperature in LTE are given in Chapters 4 and 6 (see also Vardavas and Carver 1984a).

The mean global vertical heating rate, H(z), at altitude z, can be calculated from

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