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~ -4.6Kkm or about 50% of the adiabatic value. On the basis of our stability results, we would expect no convection, and thus no con-vective heat transport. In fact the tropical atmosphere, and indeed the atmosphere as a whole, is almost always stable to dry convection; the situation is as sketched in the schematic, Fig. 4.10. We will see in Section 4.5 that it is the release of latent heat, when water vapor condenses on expansion and cooling, that leads to convective t

Temperature (K)

FIGURE 4.9. Climatological atmospheric temperature T (dashed), potential temperature 6 (solid), and moist potential temperature 6e (dotted) as a function of pressure, averaged over the tropical belt ±30°

Temperature (K)

FIGURE 4.9. Climatological atmospheric temperature T (dashed), potential temperature 6 (solid), and moist potential temperature 6e (dotted) as a function of pressure, averaged over the tropical belt ±30°

FIGURE 4.10. The atmosphere is nearly always stable to dry processes. A parcel displaced upwards (downwards) in an adiabatic process moves along a dry adiabat (the dotted line) and cools down (warms up) at a rate that is faster than that of the environment, dT£ / dz. Since the parcel always has the same pressure as the environment, it is not only colder (warmer) but also denser (lighter). The parcel therefore experiences a force pulling it back toward its reference height.

FIGURE 4.10. The atmosphere is nearly always stable to dry processes. A parcel displaced upwards (downwards) in an adiabatic process moves along a dry adiabat (the dotted line) and cools down (warms up) at a rate that is faster than that of the environment, dT£ / dz. Since the parcel always has the same pressure as the environment, it is not only colder (warmer) but also denser (lighter). The parcel therefore experiences a force pulling it back toward its reference height.

instability in the troposphere and thus to its ability to transport heat vertically. But before going on, we will introduce the very important and useful concept of potential temperature, a temperature-like variable that is conserved in adiabatic motion. This will enable us to simplify the stability condition.

### 4.3.2. Potential temperature

The nonconservation of T under adia-batic displacement makes T a less-than-ideal measure of atmospheric thermodynamics. However, we can identify a quantity called potential temperature that is conserved under adiabatic displacement.

Using the perfect gas law Eq. 1.1, our adi-abatic statement, Eq. 4-13, can be rearranged thus dp cpdT _ RT

where k = R/cp = 2/7 for a perfect diatomic gas like the atmosphere. Thus, noting that d ln x = dx/x, the last equation can be written

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