## Rt

or,notingthat p dp = dtp,

-*L = - RT = -H, d ln p g where H is the vertical scale height, Eq. 3-9. For an isothermal atmosphere (with constant scale height), z varies as ln p, which of course is just another way of saying that p varies exponentially with z (see Eqs. 3-7 and 3-8). By integrating Eq. 5-1 vertically, we see that the height of a given pressure surface is dependent on the average temperature below that surface and the surface pressure, ps, thus:

where we have set z(ps) = 0. The geopotential height of the surface is defined by Eq. 5-2, but with g replaced by its (constant) surface value. In Chapter 1 we noted that g varies very little over the depth of the lower atmosphere and so, for most meteorological purposes, the difference between actual height and geopotential height is negligible. In the mesosphere, however, at heights above 100 km, the difference may become significant (see, e.g., Problem 6 in Chapter 3).

Note that, as sketched in Fig. 5.11, low height of a pressure surface corresponds to low pressure on a z surface.

The height of the 500 mbar surface (January monthly average) is plotted in Fig. 5.12. It has an average height of 5.5 km, as we deduced in Section 3.3, but is higher in the tropics (5.88 km) than over the pole (4.98 km), sloping down from equator to pole by about 900 m.

The zonally averaged geopotential height is plotted in Fig. 5.13 as a function of pressure and latitude for mean annual conditions. Note that the difference z(p, p) - (z) (p) is plotted where (z) (p) is the horizontal average at pressure level p. Except near the surface, pressure surfaces are generally high in the tropics and low at high latitudes, especially in winter. Since surface pressure does not vary much (a few tens of mbar at most), the height of a given p

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