Upon integrating each side:
To show this consider the quantity d (ws/T) =
We can use the Clausius-Clapeyron equation to show that the first term in parentheses is much larger than the second term (recall that for every 10 K of increase in temperature there is a doubling of vapor pressure; then dws/ws ~ 1, and dT/T ~ 10/300. Compare: 1 >> 10/300). We may then substitute d(ws/T) for dws/T to a good approximation.
w s w where 0(T,p) = T (p0/p)K. This last relationship (6.80) forms an implicit functional relationship that defines a curve in the T-P plane. The relationship can also be written
The coefficient in front of the exponential, 6e, is called the equivalent potential temperature. The equivalent potential temperature is conserved along the path of a moist parcel.
For a dry adiabat 6dry (T) = constant, but for the moist adiabat, d6/dT > 0. If we solve for 6e from (6.81), we obtain
To see the physical significance of 6e let us lift the parcel until all its water is condensed out (this means p ^ 0 or z ^ to). In this limit ws ^ 0 in (6.82) and the equivalent potential temperature 6e becomes equal to the potential temperature 6. In other words, to find an equivalent potential temperature, one should lift the air parcel until all moisture is condensed and precipitates out, then compress the dry parcel adiabatically downwards until it reaches 1000 mb. The temperature the parcel attains at the 1000 hPa level is the equivalent potential temperature 6e. The whole process is supposed to occur without exchanging heat with the environment. Note that 6e is a unique label that can be attached to any air parcel, given its values of T, w and p at a particular level.
If the parcel is initially saturated and has temperature T0 at level p0, the equivalent potential temperature 6e can be calculated by substituting its temperature To, its potential temperature 6(T0,p0), and the saturation mixing ratio ws(T0) into (6.82). If the parcel is initially unsaturated, then the temperature, potential temperature, and saturation mixing ratio are to be calculated at the lifting condensation level (LCL). Since the mixing ratio w is equal to the saturation mixing ratio ws at the LCL, the formula for equivalent potential temperature for an unsaturated parcel becomes
We emphasize again that the equivalent potential temperature is conserved during both dry and moist adiabatic processes, while potential temperature is conserved only during dry adiabatic processes. This is the reason for using 6e: it can serve as a good tracer for a moving air mass. Imagine, for example, a moist flow that passes over a mountain. If air in the flow is initially unsaturated, it could be lifted by convection on the upslope side of the mountain to the LCL, where the condensation process starts. During the lifting moisture is removed by raining out on the upslope side. Then as the air descends back to the surface on the lee side of the mountain, it will be much warmer and drier than on the upslope side. This is the origin of the Chinook wind (more on this in Chapter 7). So, the temperature, potential temperature, and mixing ratio vary during both ascent and descent of air parcels. At the same time, the equivalent potential temperature is the same at the starting and ending points; it is conserved during this complicated process.
There is yet another good tracer of moist air: the so-called wet-bulb potential temperature. The wet-bulb potential temperature, 6w, is the temperature an air parcel would have if cooled from its initial state adiabatically to saturation, and then brought to 1000 hPa by a moist adiabatic process. This algorithm of finding the wet-bulb potential temperature depends on whether or not the parcel is initially saturated. If the parcel is initially saturated, it should be carried along a moist adiabat to the 1000 hPa pressure level. If the parcel is initially unsaturated, it should be lifted first to the LCL and then taken moist adiabatically to the 1000 hPa level. When descending, an air parcel may need additional water vapor to maintain saturation. The wet-bulb potential temperature, like the equivalent potential temperature, is conserved during both dry and moist adiabatic processes. So, in the case of the Chinook wind it is the same on the upslope and lee sides of the mountain.
The last useful characteristic of moist air that we introduce in this section is the saturation equivalent potential temperature 6s. Consider an unsaturated parcel. The saturation equivalent potential temperature is the equivalent potential temperature the parcel would have if it started out completely saturated. The saturation equivalent potential temperature 6s can be defined as:
It is important to understand the difference between (6.82) and (6.84). 6 and ws in (6.82) are the potential temperature and saturation mixing ratio of saturated air at temperature T, whereas the same variables in (6.84) are calculated at the temperature T of unsaturated air. The saturation equivalent potential temperature is not conserved during an unsaturated process. For saturated air, 6e is equal to 6s. The reason we introduce 6s is that it is a useful characteristic of air flow when analyzing air stability (we will discuss it briefly in Chapter 7).
6.10 Moist static energy
We can find a very simple form for the enthalpy of moist air,
where the second term represents the contribution due to the enthalpy of vaporization, and the third is the enthalpy of the vapor. If the water were to condense, the second term would contribute to raising the temperature. Neglecting the third term (which is very small compared to the others) the specific enthalpy can be written h = cpT + Lws (6.86)
where we have neglected the mass of water vapor compared to that of the dry air.
If we consider a parcel of air being lifted, there is actually another term due to the gravitational potential energy per unit mass gz, where g is the gravitational acceleration and z is the elevation above some reference level. Kinetic energy could also be added but we neglect it here. The sum of enthalpy and gravitational potential energy is conserved along a vertical path. This sum is called the moist static energy. The term static is used because we neglected kinetic energy. As the parcel is lifted, the moist static energy (call it hmse) is conserved. Note that below the LCL this means that dhmse = 0, ^ ^ = -g, (6.87)
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