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

In linear quasi-geostrophic approximation, buoyancy and vertical velocity satisfyImpermeability means $w^{\prime }=0$ at $z=-H$ and $z=0$. For a normal mode with nonzero frequency this is equivalent to the Neumann conditionswhere the primes on $\hat{\psi }$ denote derivatives with respect to $z$. Equivalently, ${\psi }_z^{\prime }=0$ at both rigid boundaries.