In planetary atmospheres, where vertical wind velocities are generally very much less than horizontal wind velocities, the assumption of hydrostatic equilibrium is a very good one. Thus, the vertical pressure difference dp across a slab of air at altitude z of density p and thickness dz subject to a gravitational acceleration g is dp =—pg dz. (4.1)
Assuming the air behaves as an ideal gas, the density may be determined from temperature T and pressure p as
where R is the molar gas constant; and m is the mean molecular weight of the atmosphere. Substituting for p in Equation (4.1) and integrating (assuming T is constant with height) we obtain
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