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

Paper 333 / 3 / a / i / Solution

{title2=QG approximation}
{wiki}

The quasi-geostrophic approximation describes slowly evolving, nearly geostrophic flow with small Rossby number. Its dynamics are governed by advection and forcing of <potential vorticity>.

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