## Cxn

From its definition, Eq. 4-17, we can see that 9 is the temperature a parcel of air would have if it were expanded or compressed adiabatically from its existing p and T to the standard pressure p0. It allows one, for example, to directly determine how the temperature of an air parcel will change as it is moved around adiabatically; if we know its 9, all we need to know at any instant is its pressure, and then Eq. 4-17 allows us to determine its temperature at that instant. For example, from the climato-logical profile shown in Fig. 4.9, a parcel of air at 300 mbar has T = 229 K (-44°C) and 9 = 323K; if the parcel were brought down to the ground (p = p0) adiabatically, thus conserving 9, its temperature would be T = 9 = 323 K (50°C).

We can express the stability of the column to dry adiabatic processes in terms of 9 as follows. Let's return to our air parcel in Fig. 4.5. At the undisturbed position z1, it has environmental temperature and pressure and therefore also environmental potential temperature 91 = 9E (z1), where 9E (z) is the environmental profile. Since the parcel preserves 9 in adiabatic motion, it still has 9 = 91 when displaced to z2. The parcel pressure is the same as that of its environment and so, from Eq. 4-17, it is warmer (or cooler) than its environment according to whether 91 is greater (or lesser) than 9E (z2). Since 9E (z2) ~ 9e(z1) + (d9/dz)E Sz = 91 + (d9/dz)E Sz, the parcel is unstable neutral stable if ' r dz

Note that Eq. 4-18 has the same form as Eq. 4-6 for an incompressible fluid, but is now expressed in terms of 9 rather than T. So another way of expressing the instability criterion is that a compressible atmosphere is unstable if potential temperature decreases with height.

Figure. 4.9 shows climatological T and 9 as functions of pressure up to 100 mbar, averaged over the tropical belt. Note that d9/dz > 0, but that dT/dz < 0. As noted earlier, we see that the climatological state of the atmosphere is stable to dry convection. However, dry convection is often observed in hot arid regions, such as deserts (e.g., the Sahara desert or Arizona) where the surface can become very hot and dry. This state of affairs is sketched in Fig. 4.11. Air parcels rise from the surface and follow a dry adiabat (conserving potential temperature) until their temperature matches that of the environment, when they will become neutrally buoyant. (In reality, the rising parcels have nonzero momentum, so they may overshoot the level of neutral buoyancy, just as observed in our laboratory convection experiment. Conversely, they may also entrain cooler air from the environment, thus reducing their buoyancy and limiting their upward penetration.) So in Fig. 4.11 (left), if the surface temperature is T1 (or T2), convection will extend to an altitude z1 (or z2). This is the atmospheric and therefore compressible analogue of the convection of an incompressible fluid (water) studied in GFD Lab II, Section 4.2.4. The analogy becomes even clearer if one views the same process in terms of 9, as sketched in Fig. 4.11 (right). The convective layer is of uniform 9 corresponding to neutral stability, just as observed in the laboratory experiment (cf. Fig. 4.6).