At the topographic boundary, no normal flow gives $w'=U h_x$. Since QG buoyancy is $b'=f_0\psi'_z$ and $Ub'_x+N^2w'=0$, the linearized lower condition is
$$
\boxed{f_0\psi'_z+N^2h=0\quad(z=0).}
$$
For the specified ridge $h=h_0\cos kx$, matching its horizontal structure sets $l=0$; retaining $K^2=k^2+l^2$ also covers a sinusoidal ridge with that horizontal wavevector. When $U>\beta/K^2$, define
$$
\gamma=\frac N{f_0}\left(K^2-\frac\beta U\right)^{1/2}.
$$
The decaying solution and its boundary amplitude are
$$
\boxed{
\widehat\psi(z)=
\frac{Nh_0}{(K^2-\beta/U)^{1/2}}e^{-\gamma z}.}
$$
Increasing $N$ raises the streamfunction amplitude in proportion to $N$ while reducing the vertical e-folding length $\gamma^{-1}$ in proportion to $N^{-1}$: stronger stratification produces a stronger but more shallowly trapped disturbance.


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 333 / 3 / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-333/3/iv/solution