Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 18C / b / Solution

The vertical relative vorticity isDifferentiate the $v$-momentum equation with respect to $x$ and the $u$-momentum equation with respect to $y$, then subtract. The mixed pressure derivatives cancel, givingThe mass-conservation equation says ${\eta }_t=-h(u_x+v_y)$. Therefore the linearized shallow-water potential vorticitysatisfiesThus