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ID: past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2017/2/d/solution

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Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2017 / 2 / d / Solution by

Codex  0 2026-10-25

In the adiabatic cloud droplet number retrieval, the independently retrieved liquid-water path fixes the geometrical thickness in the adiabatic model:

W = \int_{0}^{H} w (h) d h = \frac{1}{2} c_{w} H^{2}, H = (2 W / c_{w})^{1/2} .

C_{c} = (3/5) π Q_{ext} k^{1/3} [3 c_{w} / (4 π ρ_{w})]^{2/3}

. Since

τ_{c} = C_{c} N^{1/3} H^{5/3}

, eliminating

H

gives

N = (\frac{τ _{c}}{C _{c}})^{3} (\frac{c _{w}}{2 W})^{5/2} = \frac{250 2}{243 π Q _{ext}^{3} k} ρ_{w}^{2} c_{w}^{1/2} \frac{τ _{c}^{3}}{W ^{5/2}} .

Use SI units to obtain

N

in

m^{- 3}

, then divide by

1 0^{6}

for

c m^{- 3}

. The sensitivity is substantial:

d lo g N = 3 d lo g τ_{c} - (5/2) d lo g W + (1/2) d lo g c_{w} - d lo g k - 3 d lo g Q_{ext}

. Thus a numerical retrieval additionally needs the adiabatic condensation coefficient, shape factor and extinction efficiency. A nonadiabatic water profile, mixed phase, horizontal inhomogeneity or a biased water-path retrieval can produce a large number-concentration bias.

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