whence r dz
Note that if H(z) = H, the constant value considered in the previous section, Eq. 3-11 reduces to Eq. 3-7.
In fact, despite its simplicity, the isothermal result, Eq. 3-7, yields profiles that are a good approximation to reality. Fig. 3.6 shows the actual pressure profile for 40° N in December (corresponding to the temperature profile in Fig. 3.1) (solid), together with the profile given by Eq. 3-7, with H = 6.80 km (dashed). Agreement between the two is generally good (to some extent, the value of H was chosen to optimize this). The differences can easily be understood g p n n
from Eqs. 3-9 and 3-10. In regions where, for example, temperatures are warmer than the reference value (T0 = gH/R = 237.08 K for H = 6.80 km), such as (cf. Fig. 3.1) in the lower troposphere, near the stratopause and in the thermosphere, the observed pressure decreases less rapidly with height than predicted by the isothermal profile.
For the isothermal case, the density profile follows trivially from Eq. 3-7, by combining it with the gas law Eq. 1.1:
Thus in this case, density also falls off exponentially at the same rate as p. One consequence of Eq. 3-12 is that, as noted at the start of Chapter 1, about 80% of the mass of the atmosphere lies below an altitude of 10 km.
For the nonisothermal atmosphere with temperature T(z), it follows from Eq. 3-11 and the equation of state that
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