For
$$
\psi'=A(x,y,t)\cos\frac{n\pi z}{H},
$$
the surface value is simply $\psi'(0)=A$. The dynamic free-surface condition from part i gives
$$
\boxed{
\eta(x,y,t)=\frac{f_0}{g}A(x,y,t)}.
$$
The cosine is the leading small-surface-displacement approximation to the exact <quasi-geostrophic vertical mode>; its derivative vanishes at both rigid-boundary locations.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 333 / 3 / b / ii / Solution

{title2=QG vertical mode}

A quasi-geostrophic vertical mode separates horizontal and vertical dependence of the streamfunction. With rigid horizontal boundaries and constant stratification, the vertical eigenfunctions are $\cos(n\pi z/H)$.


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