Solution
= Solution
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.