Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 4 / b

Solution

Use a single-layer greenhouse model whose atmosphere is transparent to incident stellar light and has infrared emissivity

ϵ

. Let

T_{s}

and

T_{a}

ϵ σ T_{s}^{4} = 2 ϵ σ T_{a}^{4}, T_{a}^{4} = \frac{1}{2} T_{s}^{4} .

At the top of the atmosphere, the escaping flux is the directly transmitted surface radiation plus upward atmospheric emission:

σ T_{eq}^{4} = (1 - ϵ) σ T_{s}^{4} + ϵ σ T_{a}^{4} .

Therefore

T_{s} = \frac{T _{eq}}{( 1 - ϵ /2 ) ^{1/4}} .

For a perfectly infrared-opaque one-layer atmosphere,

T_{s} = 2^{1/4} T_{eq}

; for

ϵ = 0

T_{s} = T_{eq}

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