= Solution
Use a one-layer, hydrostatic, homogeneous fluid with reference depth $H_0$, no forcing or friction, small <Rossby number> $\epsilon=U/(|f_0|L)\ll1$, slow evolution, and leading <geostrophic balance>. For the <beta plane>, take $\beta L/|f_0|=O(\epsilon)$; the relative free-surface displacement is also small, $\eta/H_0=O(\epsilon)$, with the <Rossby deformation radius> retained at the chosen horizontal scale. This excludes the equatorial limit. If a bottom elevation $h_b$ is included, additionally require $h_b/H_0=O(\epsilon)$ and stationary topography. Set $h_b=0$ for a flat bottom.
The <quasi-geostrophic streamfunction> satisfies $\psi=g\eta/f_0$, $u=-\psi_y$, $v=\psi_x$, so <relative vorticity> is $\zeta=\nabla_h^2\psi$. Expanding the <shallow-water potential vorticity> using actual depth $H_0+\eta-h_b$ gives
$$
q=\frac{f_0}{H_0}+\frac1{H_0}\left[\nabla_h^2\psi+\beta y-\frac{f_0\eta}{H_0}+\frac{f_0h_b}{H_0}\right]+O(\epsilon^2|f_0|/H_0).
$$
Consequently the depth-scaled anomaly, the <shallow-water quasi-geostrophic potential vorticity>, is
$$
\boxed{Q=\nabla_h^2\psi-R_D^{-2}\psi+\beta y+\frac{f_0h_b}{H_0},\qquad R_D^2=\frac{gH_0}{f_0^2}.}
$$
The leading material derivative uses the <geostrophic flow>. Define $J(a,b)=a_xb_y-a_yb_x$; then conservation yields $Q_t+J(\psi,Q)=0$. The term $-\psi/R_D^2$ represents free-surface stretching, $\beta y$ the variation of planetary rotation, and $f_0h_b/H_0$ topographic stretching. Under a <rigid lid> or at scales much smaller than the <Rossby deformation radius>, the free-surface term is absent or negligible and the flat-bottom <Barotropic Rossby wave> model has $Q=\nabla_h^2\psi+\beta y$.
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