1 + 2 1 lnh4sh • ^nh4sh mcpTR (xnh3 + xh2s) 4 4 )
where xNH3 is the saturated partial pressure of ammonia above solid NH4SH particles; xH2S is the saturated partial pressure of hydrogen sulfide above the same particles; the molar heat of formation LNh4SH = 46,025.0 J/mol; and 5NH( SH is a constant of the reaction rate equation (see Section 4.3.3) whose value is 10,833.6 K-1.
At higher levels in the atmosphere, the opacity of the overlying air becomes progressively smaller and thus radiation becomes more efficient than convection at transporting heat. Hence, at these levels the temperature profile is determined by radiative equilibrium. Consider a very thin layer of opacity e high in the atmosphere with negligible atmospheric opacity above it. The atmosphere below this layer effectively emits as a black body of temperature equal to the bolometric temperature T4. Hence, the heating rate of the thin layer per unit area is simply eT4. However, heat from this layer may be emitted both upwards and downwards and thus in equilibrium; assuming Kirchoff's law, we find eaT 4 = 2e<rT4s. (4.12)
Thus the limiting temperature of the thin slab, known as the stratospheric temperature, is given by
With more detailed analysis it can be shown that the temperature in the upper atmosphere should tend gradually to this stratospheric temperature via the Milne-Eddington equation (Atreya, 1986)
where r is the IR optical thickness between the layer and space, known as the IR optical depth. Calculating T(r) from Equation (4.14) we find that the rate of increase of temperature with depth as we go down through the atmosphere is initially small and much less than the DALR. Hence, the upper atmosphere is convectively stable since parcels displaced vertically will tend to return to their original altitudes. However, since r increases quickly with depth in the atmosphere, the radiative equilibrium vertical temperature gradient increases rapidly until at some point it exceeds the dry adiabatic lapse rate. Such a temperature gradient is highly unstable and convection in the atmosphere quickly reduces the temperature gradient to the DALR. The boundary between the convective and radiative regions is known as the radiative-convective boundary.
According to Equation (4.14), the temperature in the upper atmosphere should tend to a constant value in the absence of other sources of heat. However, when we look at the temperature profiles of the planets it is found that the temperatures decrease with height to roughly the stratospheric temperature at a certain level known as the tropopause (after the Greek words for "turning" and "stop") and then increase again. The region above the tropopause, where temperature increases with height, is very stable to convection and is known as the stratosphere, since the air forms stably stratified layers. The region between the tropopause and the radiative-convective boundary is known as the upper troposphere. This is actually a slight misnomer since the lapse rate in this region is less than the DALR and the region is thus also stable to convective overturning, although turbulent overturning is important. [NB: Eddy mixing, discussed in Section 4.2 is found to be a minimum at the tropopause so this is perhaps not such a misleading word after all!] Unforced convective overturning only occurs in the atmosphere below the radiative-convective boundary, in the lower troposphere.
The increase of temperature with height in the stratosphere implies the presence of additional energy sources. These sources include
(1) absorption of ultraviolet radiation from the Sun via gaseous photodissociation reactions;
(2) absorption of sunlight by stratospheric aerosols; and
(3) absorption of near-IR sunlight by methane gas absorption bands.
To achieve thermal balance these sources of energy must be transported or radiated away. The stratospheres of all the giant planets appear to be close to radiative equilibrium and the cooling of the lower stratosphere appears to be due mainly to the thermal emission from ethane and acetylene molecules, with the upper stratosphere cooled by methane emission in the case of Jupiter and Saturn (Yelle et
al., 2001). Acetylene and ethane are photochemical products derived from the photolysis of methane and are observed in the stratospheres of all the giant planets. Thus, the temperature structure in the stratospheres of the giant planets depends critically on the vertical distribution of photochemical products in the same way that the stratospheric temperature profile of the Earth depends critically on the abundance of another photochemical product, ozone. The stratospheres of Mars and Venus, which do not contain significant quantities of photochemical products such as ozone, do not have nearly so well a defined tropopause. Hence, in a peculiar way the terrestrial stratosphere has, in this sense, more in common with the stratospheres of the giant planets than it does with the stratospheres of the other terrestrial planets (Yelle et al., 2001)! From Figure 4.1 it can be seen that the stratospheric temperatures of all the giant planets are rather similar, which is puzzling considering that the solar flux at Neptune is 33 times smaller than at Jupiter (Table 3.1). Chamberlain and Hunten (1987) note that this may be due to the positioning of the main ethane and acetylene absorption bands with respect to the Planck function at the stratospheric temperature. Although these gases are not very abundant in the stratosphere, their absorption bands overlap with the Planck function much better than the main methane band at 7.7 ^m and this is why stratospheric temperatures depend so critically on their abundances. At 150 K, the Planck function peaks at 19 ^m, and only slightly overlaps with the main acetylene band at 13.7 ^m and even less with
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