Adiabatic Atmosphere

The lower 10–15 km of the atmosphere, the troposphere, is often in a convective steady state with constant entropy, not constant temperature (PV^γ is independent of the altitude, where γ = C_P/C_V).

a) Find the change of temperature in this model with altitude dT/dz.

b) Estimate dT/dz in K/km. Consider the average diatomic molecule of air with molar mass μ = 29 g/mole.

SOLUTION

a) Starting from the ideal gas law, we can express the temperature T as a function of pressure P and the mass density ρ:

(1)

where P and ρ are functions of the height z above the surface of the Earth: P ≡ P(z), P ≡ ρ(z). Taking the derivative of T with respect to z, we have

(2)

We need to express dρ in terms of dP. The fact that PV^γ is independent of altitude allows us to write

where B is some constant. So

(3)

Substituting (3) into (2), we obtain

(4)

Assuming that the acceleration of gravity is constant, using the hydrostatic pressure formula

(5)

and substituting (5) into (4), we can write

(6)

b) For the atmosphere, using diatomic molecules with C_V = 5R/2 and C_P = 7R/2, we have from (6),

This value of |dT/dz| is about a factor of 2 larger than that for the actual atmosphere.