In linear <quasi-geostrophic approximation>, buoyancy and vertical velocity satisfy
$$
f_0\psi'_{zt}+N^2w'=0.
$$
Impermeability means $w'=0$ at $z=-H$ and $z=0$. For a normal mode with nonzero frequency this is equivalent to the Neumann conditions
$$
\boxed{
\hat\psi'(-H)=0,\qquad
\hat\psi'(0)=0},
$$
where the primes on $\hat\psi$ denote derivatives with respect to $z$. Equivalently, $\psi'_z=0$ at both rigid boundaries.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 333 / 3 / a / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-333/3/a/i/solution