In the convective equilibrium of the atmosphere, the air is supposed to change in density and pressure without exchange of heat by conduction; and then
ρ/ρ0 = (θ/θ0)n, p/p0 = (θ/θ0)n + 1,
(11)
| dz | = | 1 | dp | = (n + 1) | p | = (n + 1) R, γ = 1 + | 1 | , | |
| dθ | ρ | dθ | ρθ | n |
where γ is the ratio of the specific heat at constant pressure and constant volume.
In the more general case of the convective equilibrium of a spherical atmosphere surrounding the earth, of radius a,
| dp | = (n + 1) | p0 | dθ | = − | a2 | dr, | |
| ρ | ρ0 | θ0 | r2 |
(12)
gravity varying inversely as the square of the distance r from the centre; so that, k = p0/ρ0, denoting the height of the homogeneous atmosphere at the surface, θ is given by
(n + 1) k (1 − θ/θ0) = a(1 − a/r),