= Solution
The first-order perturbation of the <vortensity> is
$$
\boxed{f'=
\frac1\Sigma(\partial_xv_y-\partial_yv_x)
-\frac{2\Omega-S}{\Sigma^2}\Sigma'}.
$$
Taking the curl of the linearized momentum equation and using the linearized continuity equation yields
$$
\boxed{\frac{Df'}{Dt}=0}.
$$
This is <linearized vortensity conservation>. It is the perturbative form of material conservation of <potential vorticity> in an inviscid <barotropic fluid>; the forcing $-\nabla\Psi$ is a gradient and therefore creates no <vorticity>.
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