The first law, Eq. 4-11 can then be written:

= cp dT--, p where cp = R + cv is the specific heat at constant pressure.

For adiabatic motions, SQ = 0, whence

Now if the environment is in hydrostatic balance then, from Eq. 3-3, dp = -gpE dz, where pE is the density of the environment (since the parcel and environmental pressures must be locally equal). Before being perturbed, the parcel's density was equal to that of the environment. If the displacement of the parcel is sufficiently small, its density is still almost equal to that of the environment, p ~ pE, and so under adiabatic displacement the parcel's temperature will change according to

where rd is known as the dry adiabatic lapse rate, the rate at which the parcel's temperature decreases with height under adiabatic displacement. Given cp = 1005 J kg-1K-1 (Table 1.4), we find rd ~ 10K km-1.

To determine whether the parcel experiences a restoring force on being displaced from z1 to z2 in Fig. 4.5, we must compare its density to that of the environment. At z2, the environment has pressure p2, temperature T2 ~ T1 + (dT/dz)E Sz, where (dT/dz)E is the environmental lapse rate, and density p2 = p2/RT2. The parcel, on the other hand, has pressure p2, temperature TP = T1 - rdSz, and density pP = p2/RTP. Therefore the parcel will be positively, neutrally, or negatively buoyant according to whether TP is greater than, equal to, or less than T2. Thus our stability condition can be written unstable ^ / ,T \ f < -rd neutral } if (dT) I = -Td . stable J \dzh\> -rd

Therefore, a compressible atmosphere is unstable if temperature decreases with height faster than the adiabatic lapse rate.

This is no longer a simple "top-heavy" criterion (as we saw in Section 3.3, atmospheric density must decrease with height under all circumstances). Because of the influence of adiabatic expansion, the temperature must decrease with height more rapidly than the finite rate rd for instability to occur.

The lower tropospheric lapse rate in the tropics is, from Fig. 4.9, dT dz

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