In fact, the temperature inside the atmosphere is not homogeneous but depends on the altitude. Within the troposphere (lowest atmospheric layer), the temperature decreases with the altitude, which can be seen from the fact that there is a snow cap on high mountains even in tropical regions.
A simple reason for this is the fact that gases cool down on expansion, when they can not exchange heat with their environment. When an air parcel rises in the atmosphere, the pressure decreases, causing the air to expand.
Calculate the temperature decrease of an air package with the height near sea level under the following assumptions:
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The mass density ρ of the air parcel is inversely proportional to the volume V : ρ = V M n ∝ V 1 Therefore, the adiabatic condition P V κ = const can be rewritten to ⇒ ρ κ P ρ = const = const ⋅ P 1 / κ = ρ 0 exp [ − κ H z ] ∣ P ∝ e − z / H with the density ρ 0 = ρ ( z = 0 ) at sea level. Here we have used the barometric height formula from the last section. Thus, the density also depends exponentially on the height, but drops faster than the pressure P by the factor κ = 1 . 4 . Since we know pressure and density as a function of altitude, we can also use the general gas equation to determine the temperature curve: ⇒ P T = const ⋅ ρ T = const ⋅ ρ P = T 0 exp [ − κ κ − 1 H z ] ∣ P ∝ e − z / H , ρ ∝ e − z / κ H with the temperature T 0 = T ( z = 0 ) . At sea level, therefore, the temperature gradient gives the result Γ = − d z d T ∣ ∣ ∣ ∣ z = 0 = κ κ − 1 H T 0 = 1 . 4 1 . 4 − 1 8 4 1 7 m 2 8 8 K ≈ 9 . 8 ⋅ 1 0 − 3 m K The quantity Γ is also known as the dry adiabatic lapse rate.