Differentiate the second momentum equation in $x$, the first in $y$, subtract, and use continuity:
$$
(v_x-u_y)_t=-f(u_x+v_y)=\frac f{c^2}\phi_t.
$$
Hence the linear <shallow-water potential vorticity>
$$
\boxed{P=v_x-u_y-\frac f{c^2}\phi}
$$
satisfies $P_t=0$.

Solution

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

 Solution

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