= Solution
Let $\mathbf u=(u,v)$ and let the layer thickness be $H=H_0+\eta$. The inviscid <shallow water equations> are
$$
\frac{D\mathbf u}{Dt}+f\widehat{\mathbf z}\times\mathbf u
=-g\nabla(\eta-h_b),
\qquad
H_t+\nabla\mathbin\cdot(H\mathbf u)=0.
$$
They follow from a homogeneous incompressible fluid with small aspect ratio, <hydrostatic pressure>, negligible vertical acceleration, horizontal velocity nearly uniform through the depth, a material free surface, and a rigid stationary bottom. Rotation may be represented by an <f-plane> or <beta plane>. The model is useful because many atmospheric and oceanic motions are horizontally much broader than their depth, while the free surface or an internal density interface still supports waves and <potential vorticity> dynamics.
Let $\zeta=v_x-u_y$. Taking the vertical curl of momentum gives
$$
\frac{D(\zeta+f)}{Dt}=-(\zeta+f)\nabla\mathbin\cdot\mathbf u,
$$
where $Df/Dt=\beta v$ supplies the planetary-vorticity term when $f$ varies. Continuity gives $DH/Dt=-H\nabla\cdot\mathbf u$. Combining the two equations yields material conservation of <shallow-water potential vorticity>:
$$
\boxed{q=\frac{\zeta+f}{H},
\qquad \frac{Dq}{Dt}=0.}
$$
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