Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 333 / 4 / c

Solution

Put

l = π / L

. For the specified quasi-geostrophic streamfunction, the complex amplitudes of the disturbance fields are

u = - l ψ cos (l y), v = ik ψ sin (l y) .

The product of these two amplitudes is purely imaginary after one is conjugated, so its zonal mean vanishes:

\overline{u^{'} v^{'}} = \frac{1}{2} Re (u v^{*}) = 0.

Consequently

\overline{F}^{(y)} = 0 .

ρ = - \frac{ρ _{0} f _{0}}{g} ψ_{z} sin (l y) .

Therefore

\overline{ρ^{'} v^{'}} = - \frac{ρ _{0} f _{0} k}{2 g} Im (ψ_{z} ψ^{*}) sin^{2} (l y) .

Since

d ρ_{s} / d z = - ρ_{0} N^{2} / g

\overline{F}^{(z)} = \frac{f _{0}^{2}}{N ^{2}} \frac{k}{2} Im (ψ_{z} ψ^{*}) sin^{2} \frac{π y}{L} .

Thus it has the stated form

\overline{F}^{(z)} = F_{0} Θ (z) sin^{2} \frac{π y}{L}, F_{0} = \frac{f _{0}^{2}}{N ^{2}},

with

Θ (z) = \frac{k}{2} Im (ψ_{z} ψ^{*}) .

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