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Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2018 / 1 / b / Solution by

Codex  0 2026-10-25

Take the convection to maintain a dry, nearly reversible adiabat, with specific gas constant

R

. Hydrostatic equilibrium and an ideal-gas equation of state will be used separately to convert pressure to height. For an adiabatic displacement, the first law of thermodynamics in enthalpy form gives

c_{p} (T) d T - \frac{d p}{ρ} = 0, \frac{A + BT}{T} d T = R \frac{d p}{p} .

(1)

The adiabat with linear heat capacity follows by integrating from the surface: this gives the monotone implicit temperature–pressure relation

A ln \frac{T}{T _{s}} + B (T - T_{s}) = R ln \frac{p}{p _{s}} .

(2)

An explicit form, using the positive real Lambert W branch, is

T (p) = \frac{A}{B} W_{0} [\frac{B T _{s}}{A} e^{B T_{s} / A} (\frac{p}{p _{s}})^{R / A}] (B > 0) .

(3)

B \to 0

T = T_{s} (p / p_{s})^{R / A}

. At fixed

p < p_{s}

\frac{\partial T}{\partial B}_{p} = \frac{T _{s} - T}{A / T + B} > 0.

(4)

Larger heat capacity therefore keeps the upper atmosphere warmer at the same pressure. Equivalently

d ln T / d ln p = R / (A + BT)

becomes smaller. The original normalized sketch below shows the resulting family of profiles; no particular gas constants or surface temperature are assumed there.

The height equations are

\frac{d p}{d z} = - ρ g = - \frac{p g}{RT}, \frac{d T}{d z} = - \frac{g}{A + BT} .

(5)

Integrating the latter gives

g z = A (T_{s} - T) + \frac{B}{2} (T_{s}^{2} - T^{2}), T (z) = \frac{( A + B T _{s} ) ^{2} - 2 B g z - A}{B} .

(6)

The physically selected root starts at

T_{s}

and decreases. Its formal upper boundary is

z_{*} = \frac{A T _{s} + B T _{s}^{2} /2}{g} .

(7)

At that height

T \to 0

and

p = p_{s} (T / T_{s})^{A / R} e^{B (T - T_{s}) / R} \to 0

. The

B = 0

limit is

z_{*} = A T_{s} / g

. This finite-height zero is a property of the assumed polytropic-like model, not a claim that a real atmosphere reaches absolute zero: molecular heat capacities, composition, transport and the continuum ideal-gas description cease to support that extrapolation.

Figure 1.
Adiabats with increasing heat capacity and the limited-temperature latent heat inferred from an Antoine fit
. Original calculations. Adiabats use dimensionless parameters $R / A = 2/7$ and $b = B T_{s} / A$ ; the latent-heat plot distinguishes the stated fit range from an invalid extrapolation toward the known critical point. **Details**.

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