Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2021 / 1 / b

Solution

P (x) = P_{d} + \frac{ϵ Δ p x}{1 + a x}, E (x) = x P (x) .

Both functions increase for

0 \leq x \leq 1

. At the endpoint,

P (1) = E (1) = P_{d} + Δ p .

Therefore the physically admissible hydrogen–water vapor mixture requires

E (T_{s}) < P_{d} + Δ p .

Beyond this bound, the formal positive quadratic equation root requires

x > 1

, equivalently a negative hydrogen partial pressure. It is the physical constraints, rather than loss of real roots, that end the solution branch. For a concrete example, put

P_{d} = Δ p = 1

in consistent pressure units,

ϵ = 9

, and

E = 3

. The positive root of

P^{2} + 23 P - 51 = 0

is about

2.037

x ≃ 1.473

, which is impossible.

The limiting temperature is obtained from

E (T_{*}) = P_{d} + Δ p

\frac{1}{T _{*}} = \frac{1}{T _{t}} - \frac{R _{v}}{L _{v}} ln \frac{P _{d} + Δ p}{E _{t}} .

This finite limit exists above

T_{t}

E_{t} < P_{d} + Δ p < E_{t} exp [L_{v} / (R_{v} T_{t})]

. The upper qualification matters mathematically: the stipulated constant-latent heat formula has a finite limiting saturation pressure as

T \to \infty

, so for an arbitrarily large unspecified

P_{d} + Δ p

the printed assertion of a finite breakdown temperature is not unconditional. For a concrete formal counterexample, choose

P_{d} + Δ p = 1 0^{15} Pa

: with the usual triple-point constants the model has

E (\infty) ≃ 2.5 \times 1 0^{11} Pa

, so it never reaches the proposed inventory limit. This is a counterexample inside the stipulated equations, not a physically credible high-pressure ideal gas ocean. Actual liquid water also ceases to exist above its critical point, before such an extrapolation would make sense.

Physically, the imposed finite-pressure-thickness layer can hold no more than

Δ p / g

x \to 1

, it expels all its hydrogen into the dry upper layer. A still larger surface saturation pressure cannot coexist with this geometry. The layer must deepen, the temperature profile must change, or the assumed phase equilibrium must fail. If