C LIMATE AND A TMOSPHERIC T HERMODYNAMICS
3.6 Lapse Rate
It is possible to express the static temperature (Ts) as a function of the total temperature (Tt), as in equation (30). Total temperature is also known as the stagnation temperature and may be expressed in terms of Mach number and ratio of the specific heats at constant pressure to that of constant volume ( cp/cV); Equation (31) presents the pressure () and temperature () ratios. In practice, the total temperature is measured by inserting a probe in the vicinity of the aircraft where the relative speed of the flow surrounding the aircraft (the difference between the airflow and aircraft speed) becomes zero. In reality, however, this relative velocity may not become zero, and therefore a correction factor (less than 1) is included in the calculations (e).
1 2
1 2
t s
T e M
T
(30)
Ts/T0, Ps/P0 (31)
and is directly dependent on the absolute temperature—the higher the pressure, the higher the temperature is required and vice versa. The rate of phase change remains high at the beginning of the process; however, it is reduced as more water is transformed to gas in the form of vapor.
This is the reason why the saturated lapse rate (change of temperature with respect to altitude for the saturated air parcel) is smaller than the lapse rate of dry air. More energy in the form of heat is required to cool down a parcel of air that includes a water vapor component due to the presence of contaminant nuclei and water vapor; a dry parcel of air can cool faster given its purity and lack of contamination. As a parcel of air is elevated, assuming that it starts from the dry state, it cools at a lapse rate of 3 °C per 1,000 ft (10 °C per 1 km) until reaching the saturation state.
The elevation at which the saturation state is achieved depends on the air parcel’s initial temperature. The parcel is then cooled at a saturated adiabatic rate (1.5 °C per 1,000 ft, 4.9 °C/km) in an adiabatic-reversible process (Gw). The moving air facilitates this cooling process, moving the air particles to higher elevations. The saturation continues until the formation of liquid from vapor is completed under a latent heat that is negatively related to temperature (rain stage). The cooling continues down to the freezing point (0 °C), where the hail stage, which is an isothermal process, commences; the rate of cooling during this stage remains constant, since the latent heat of freezing is independent of temperature. This stage continues until the phase change from liquid to solid is completed. The parcel of air is still ascending with a reduced lapse rate due to the release of the latent heat of evaporation during the rain stage. Even after the majority of the liquid has transformed into a solid in the form of ice, there are still a number of vapor particles that are deposited into a solid in the form of ice crystals—the snow stage. The ascent of the air parcel continues until all vapor transforms into ice and is done in a dry adiabatic rate that is larger than that of the rain stage. Since the opposite process follows the same cooling rate and temperature profile in a majority of cases, it is called a reversible adiabatic process (Figure 8).
In a semi-adiabatic process, the formation of the precipitation happens at the moment of completion of each phase, starting with rain and then developing into snow. Hail does not form during this process since there is no liquid left in the air parcel after the completion of the rain stage to form ice. This process is irreversible. After completion of the snow stage, where precipitation in the form of snow has left the air parcel, meaning that
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there is no remaining moisture, the parcel of dry air therefore returns to its original state, which happens as a non-reversible adiabatic process and with a dry lapse rate. The temperature of the final state is higher than that of the initial one, and this final state temperature is known as the equivalent temperature.
Equation (33) presents the relationship between moist air properties and the change in the ratio of the water content to that of the dry air (dws) for an adiabatic or semi-adiabatic process, where Lv is the latent heat of vaporization. dws depends on the water content.
1 1 v s
P
dT dP L
T P TC dw
(33)
The temperature gradient between the atmospheric layers is often used to calculate the spread (i.e., the difference between the dry-bulb and dew- point temperatures) as well as the lapse rate as a function of the altitude.
These two values are employed to calculate the saturation and freezing levels required to determine cloud base and icing. The levels (ft) are calculated by dividing the spread (°C) by the lapse rate for the given altitude (°C/ft). Note that the standard lapse rate set by ICAO is 1.98 °C per 1,000 ft (6.6 °C/km).
Given the level of water vapor in the air parcel, there are three different types of lapse rate: dry, adiabatic, and saturated (moist) (Table 1).
TABLE 1 Dry, adiabatic, and saturated (moist) lapse rates.
FIGURE 8 Cooling stages in the atmosphere versus the temperature.
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