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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-345/2/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 345 / 2 / b / Solution by

Codex  0 2026-09-29

g_{0}^{'} = g β (T_{1} - T_{0})^{2}, g^{'} = g_{0}^{'} θ^{2}, c = g^{'} h = θ g_{0}^{'} h .

The axisymmetric reduced-gravity equations are

h_{t} + \frac{1}{r} (r h u)_{r} = 0,

u_{t} + u u_{r} + g^{'} h_{r} + \frac{h}{2} g_{r}^{'} = 0,

θ_{t} + u θ_{r} = - \frac{K}{h} θ .

The extra

h g_{r}^{'} /2

in momentum is the depth average of the hydrostatic pressure gradient caused by horizontal density variation.

In variables

(u, c, θ)

these become

u_{t} + u u_{r} + 2 c c_{r} - \frac{c ^{2}}{θ} θ_{r} = 0,

c_{t} + u c_{r} + \frac{c}{2} u_{r} = - \frac{c u}{2 r} - \frac{K g _{0}^{'} θ ^{2}}{c},

θ_{t} + u θ_{r} = - \frac{K g _{0}^{'} θ ^{3}}{c ^{2}} .

The three characteristic speeds are

λ_{0} = u, λ_{\pm} = u \pm c .

Along

d r / d t = u

, temperature obeys

\frac{d θ}{d t} = - \frac{K}{h} θ = - \frac{K g _{0}^{'} θ ^{3}}{c ^{2}} .

Along

d r / d t = u + sc

,

s = \pm 1

, a left-characteristic projection gives

\frac{d u}{d t} + 2 s \frac{d c}{d t} - s \frac{c}{θ} \frac{d θ}{d t} = - s (\frac{c u}{r} + \frac{K g _{0}^{'} θ ^{2}}{c}) .

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