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ID: past-exam-of-the-atmospheric-physics-course-of-the-university-of-cambridge/2014/paper-1/1/c/solution

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Past exam of the atmospheric physics course of the University of Cambridge / 2014 / Paper 1 / 1 / c / Solution by

Codex  0 2026-10-20

The cloud heating rate in this band is zero: conservative scattering changes photon directions but transfers no net energy to the cloud matter. Let

F_{c} = 9.5 e^{- 0.02/0.5} = 9.1275 W m^{- 2}

enter its top and

t_{c} = e^{- 20}

. In the stipulated single-scattering escape approximation, the downward and upward outgoing fractions are

D = t_{c} + (0.75 + \frac{0.25}{2}) (1 - t_{c}), U = \frac{0.25}{2} (1 - t_{c}), D + U = 1.

(1)

Their fluxes are approximately

7.9866

and

1.1409 W m^{- 2}

. With upward-positive net flux, both cloud boundaries have

F_{net} = - D F_{c}

; hence

\overline{\dot{T}} = \frac{F _{net} ( z _{b} ) - F _{net} ( z _{t} )}{ρ c _{v} Δ z} = 0 K s^{- 1} .

(2)

The layer heat capacity is

ρ c_{v} Δ z = 3.30 \times 1 0^{5} J m^{- 2} K^{- 1}

. Absorption outside the cloud heats those other layers. Dropping the upward flux would spuriously turn reflection into cloud heating. Neglecting multiple scattering is quantitatively poor at optical depth ten, but conservative multiple scattering still cannot create material heating in this band.

Figure 1.
Conservative single-scattering redistribution balances incoming and outgoing cloud-band energy
. **Source**. **Details**.

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