 Solution

ID: past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2021/3/a/solution

Before condensation, assume a closed parcel with fixed atmospheric mixing ratio, constant specific heat capacities, and negligible irreversible heating. The first law of thermodynamics for a dry adiabatic process is

c_{p} d T - \frac{1}{ρ} d p = 0.

The parcel remains in approximate pressure equilibrium with a hydrostatic environment. With the standard approximation that parcel and environmental densities are close,

d p / d z ≃ - ρ g

, hence

\frac{d T}{d z} = - \frac{g}{c _{p}} = - Γ_{d}, T (z) = T_{s} - Γ_{d} z .

9.8 K k m^{- 1}

for air. More precisely the environmental hydrostatic balance gives

d T / d z = - g ρ_{env} / (c_{p} ρ_{parcel})

; the familiar constant dry adiabatic lapse rate uses their ratio close to one.

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