where Cp is the molar heat capacity at constant pressure of the parcel; S is the entropy of the parcel; and where the second partial derivative in Equation (4.4) has been replaced using one of Maxwell's relations (e.g., Finn, 1993). Assuming the gas is ideal,
we may calculate dV/dT from the ideal gas equation pV = RT and derive dT R
~T = ~p where R is again the molar gas constant. Substituting for dp from the hydrostatic equation (Equation 4.1) we find dT
dz where m is the mean molecular weight of the atmosphere and thus dT m a ^
where cp is the specific heat capacity at constant pressure (J kg-1 K-1); and Yd is called the dry adiabatic lapse rate (DALR).
In parts of the atmosphere where the air contains volatiles, cooling caused by upward motion of air parcels causes the condensation of cloud particles and the subsequent release of latent heat, which causes the temperature to drop more slowly with height than expected for a dry atmosphere. For air at temperature T, where several gas components are saturated, the saturated adiabatic lapse rate (SALR) may be shown to be (Andrews, 2000; Atreya, 1986)
where Li is the molar latent heat of vaporization (or sublimation) of the i th condensing component; R is the molar gas constant; m is the mean molecular weight of the air (excluding condensates); and xi is the saturated volume mixing ratio (or mole fraction) of the i th component defined as xi = pi /p, where p is the total pressure and pi is the saturation vapor pressure of the constituent at the local temperature. In the terrestrial atmosphere, Ts is usually significantly less than Td (e.g., for the Earth, rs = 6K/km to 9K/km, while Td ~ 9.8K/km). For Jupiter and Saturn, the difference is usually small since the volume mixing ratios of condensing species is small. However, for Uranus and Neptune the abundance of condensing species is much higher and thus significant differences between Td and Ts exist at levels of major cloud formation. A major difference between giant planet atmospheres and the terrestrial atmosphere is that condensing species in the giant planet atmospheres (H20, NH3, CH4, etc.) are all heavier than the bulk of the hydrogen-helium air. The opposite is true for the Earth's atmosphere where water vapor is lighter than the bulk of the nitrogen-oxygen atmosphere. As a result "moist" air is naturally buoyant in the terrestrial atmosphere and tends to rise, whereas "moist" air is naturally dense in giant planet atmospheres and tends to sink. Hence, while clouds may form in the Earth's atmosphere simply by moist, buoyant air rising to its condensation level, the formation of clouds in the giant planet atmospheres is more complicated. Another difference is that cp, and thus Td, can vary significantly with height, especially at cold temperatures where, as we will see in Section 4.1.3, the heat capacity is a strong function of temperature and of the ortho:para ratio of molecular hydrogen. This effect is difficult to observe in the warmer tropospheres of Jupiter and Saturn, but is significant in the cooler atmospheres of Uranus and Neptune. These latter planets have the additional complication that methane, which has a significant abundance, freezes out at roughly the 1 bar level. This leads to significant changes in both cp and the mean molecular weight of the atmosphere at this level.
In addition to single condensate clouds, two-component clouds may also form in the atmospheres of the giant planets, of which the most important is probably solid ammonium hydrosulfide NH4SH, which may form by the reaction
The formation of the cloud releases additional heat that affects the lapse rate, and it may be shown that the equation for the saturated adiabatic lapse rate must be modified to (Atreya, 1986)
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