From the definition of potential temperature, Eq. 4-17, ln 6 = ln T — kln p + constant, and so this can be written d ln 6 = —^dq — -d(~~T

cpT \CpT

where we have made the approximation of slipping the factor L/cpT inside the derivative, on the grounds that the fractional change in temperature is much less than that of specific humidity (this approximation is explored in Problem 6 at the end of the chapter). Then we may conveniently define equivalent potential temperature to be

6e cpT

such that d6e = 0 in adiabatic processes. The utility of 6e is that:

1. It is conserved in both dry and wet adiabatic processes. (In the absence of condensation, q is conserved; hence both 6 and 6e are conserved. If condensation occurs, q — qt. (p,T); then 6e is conserved but 6 is not.)

2. If the air is dry, it reduces to dry potential temperature (6e — 6 when q - 0).

3. Vertical gradients of 6e tend to be mixed away by moist convection, just like the T gradient in GFD Lab II.

This last point is vividly illustrated in Fig. 4.9, where climatological vertical profiles of T, 6 and 6e are plotted, averaged over the tropical belt ± 30° (see also Fig. 5.9). Note how 6 increases with height, indicating that the tropical atmosphere is stable to dry convection. By contrast, the gradient of 6e is weak, evidence that moist convection effectively returns the tropical atmosphere to a state that is close to neutral with respect to moist processes (d6e/dp — 0).

Having derived conditions for convective instability of a moist atmosphere, let us now review the kinds of convection we observe in the atmosphere and their geographical distribution.

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