Solution
= Solution
At the displaced <free surface>, constant atmospheric pressure and the <hydrostatic approximation> give the dynamic condition
$$
\eta=\frac{f_0}{g}\psi\qquad\text{at }z=0.
$$
The linearized kinematic condition is $w=D_g\eta/Dt$. Combining these relations with
$$
w=-\frac{D_g}{Dt}\left(\frac{f_0\psi_z}{N^2}\right)
$$
gives
$$
\boxed{
\frac{D_g\psi_z}{Dt}+\frac{N^2}{g}\frac{D_g\psi}{Dt}=0
}\qquad\text{at }z=0.
$$