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
$$
\boxed{
\frac{D_g\psi_z}{Dt}+\frac{N^2}{g}\frac{D_g\psi}{Dt}=0
}\qquad\text{at }z=0.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 333 / 3 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-333/3/i/solution