Using the equation of state of air, Eq. 1-1, we may rewrite Eq. 3-3 as dp = _ gp_ RT

In general, this has not helped, since we have replaced the two unknowns, p and p, by p and T. However, unlike p and p, which vary by many orders of magnitude from the surface to, say, 100 km altitude, the variation of T is much less. In the profile in Fig. 3.1, for example, T lies in the range 200-280 K, thus varying by no more than 15% from a value of 240 K. So for the present purpose, we may replace T by a typical mean value to get a feel for how p and p vary.

3.3.1. Isothermal atmosphere

where H, the scale height, is a constant (neglecting, as noted in Chapter 1, the small dependence of g on z) with the value

RTn g

If H is constant, the solution for p is, noting that by definition, p = ps at the surface z = 0, p(z) = ps exp( - -H) . (3-7)

Alternatively, by taking the logarithm of both sides we may write z in terms of p thus:

Thus pressure decreases exponentially with height, with e-folding height H. For the troposphere, if we choose a representative value T0 = 250 K, then H = 7.31 km. Therefore, for example, in such an atmosphere p is 100 hPa, or one tenth of surface pressure, at a height of z = H x (ln 10) = 16.83 km. This is quite close to the observed height of the 100 hPa surface. Note, very roughly, the 300 hPa surface is at a height of about 9 km and the 500 hPa surface at a height of about 5.5 km.

3.3.2. Non-isothermal atmosphere

What happens if T is not constant? In this case we can still define a local scale height such that

where H(z) is the local scale height. Therefore

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