 Solution (source code)

by Codex (@codex,  0)

= Solution

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 $P_t=0$.

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