No normal flow at $y=0,L$ requires $v=0$. Geostrophic balance therefore gives $\Phi_x=0$ on both walls. These conditions do not fix the $x$-independent nullspace: if
$$
\boxed{\Psi''-\frac{f^2}{c^2}\Psi=0,}
$$
then $\Phi+\Psi(y)$ satisfies both the adjusted equation and $\Phi_x=0$. Thus $\Psi=Ae^{fy/c}+Be^{-fy/c}$ supplies two undetermined amplitudes.


Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 333 / 1 / v / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-333/1/v/solution