= Solution
For the <shallow-water approximation>, let the vertical scale be much smaller than the horizontal scale, so vertical acceleration is negligible and <pressure> is hydrostatic. Assume constant <density>, a flat impermeable bottom, a <free surface> under constant atmospheric <pressure>, and a horizontally moving column with no leading-order vertical shear. Neglect friction. Taking the bottom at $z=0$, <hydrostatic pressure> is $p=p_a+\rho g(h-z)$, so its horizontal gradient is independent of $z$. The horizontal momentum equations are therefore
$$
\frac{D_Hu}{Dt}=-gh_x,\qquad \frac{D_Hv}{Dt}=-gh_y,\qquad \frac{D_H}{Dt}=\partial_t+u\partial_x+v\partial_y.
$$
Integrating <incompressibility> from the bottom to the <free surface>, using zero bottom vertical <velocity> and the surface kinematic condition $w=h_t+uh_x+vh_y$, gives
$$
\boxed{h_t+\partial_x(hu)+\partial_y(hv)=0.}
$$
Together these are the nonrotating <shallow water equations>.
Rotation with angular <velocity> $f\hat{\boldsymbol z}/2$ adds <Coriolis force> terms $-fv$ and $+fu$ to the left sides of the $u$ and $v$ equations, respectively. The centrifugal potential is absorbed into the background <pressure> and effective gravity. Thus
$$
D_Hu-fv=-gh_x,\qquad D_Hv+fu=-gh_y.
$$
Write $\zeta=v_x-u_y$ and $d=u_x+v_y$. Taking the horizontal curl, including the derivatives of the advecting <velocity>, yields
$$
D_H\zeta=-(\zeta+f)d.
$$
For example the curl of the <Coriolis force> contribution is $fd$, and the curl of nonlinear <advection> supplies $\zeta d$. Since $q_a=f+\zeta$ and $D_Hh=-hd$, division gives
$$
\boxed{\frac{D_H}{Dt}\left(\frac{q_a}{h}\right)=\frac{-q_a d}{h}+\frac{q_a hd}{h^2}=0.}
$$
This is exact <shallow-water potential vorticity> conservation within the rotating <shallow water equations>.
For the <quasi-geostrophic approximation>, take a horizontal scale $L$, speed $U$, mean depth $H$, and slow time $L/U$. Require small <Rossby number> $\mathrm{Ro}=U/(|f|L)\ll1$ and small surface departure $\eta=h-H$. Introduce the <Rossby deformation radius> $L_R=\sqrt{gH}/|f|$ and <Burger number> $\mathrm{Bu}=L_R^2/L^2=O(1)$. <Geostrophic balance> gives $g\eta/L\sim|f|U$, hence $\eta/H\sim\mathrm{Ro}/\mathrm{Bu}\ll1$. The <Froude number> is $U/\sqrt{gH}=\mathrm{Ro}/\sqrt{\mathrm{Bu}}\ll1$. These assumptions select slow balanced motion, with fast gravity-wave transients absent at leading order.
Define the <quasi-geostrophic streamfunction> by $\psi=g\eta/f$. The leading <geostrophic flow> is $(u_g,v_g)=(-\psi_y,\psi_x)$, which is horizontally nondivergent. Expanding the exact <shallow-water potential vorticity> gives
$$
\frac{f+\zeta}{H+\eta}=\frac{f}{H}+\frac{1}{H}\left(\nabla_H^2\psi-\frac{\psi}{L_R^2}\right)+\text{higher-order terms}.
$$
The correction is smaller than $f/H$ by $O(\mathrm{Ro})$. Advecting it by the leading <geostrophic flow>, while the constant leading term has zero gradient, gives
$$
\boxed{Q=f+\nabla_H^2\psi-L_R^{-2}\psi,\qquad Q_t+J_{xy}(\psi,Q)=0,\qquad J_{xy}(A,B)=A_xB_y-A_yB_x.}
$$
Here <potential-vorticity conservation> means that $Q$ is carried unchanged by the leading horizontal trajectories. <Potential-vorticity inversion> means solving the elliptic equation $(\nabla_H^2-L_R^{-2})\psi=Q-f$, with appropriate boundary or decay conditions, to recover $\psi$ and thus the <velocity>. Conservation evolves the scalar field; inversion obtains its <velocity> field.
For a mean current $U(y)=-\bar\psi_y$, substitute $Q=\bar Q+Q'$ and $\psi=\bar\psi+\psi'$ into <potential-vorticity conservation>. Retaining first-order terms gives
$$
\boxed{Q'_t+UQ'_x+\bar Q_y\psi'_x=0.}
$$
Let $\Delta Q=\bar Q(0^+)-\bar Q(0^-)>0$. Differentiating the mean inversion relation gives $\bar Q_y=-U''+L_R^{-2}U$. Away from $y=0$ this is zero, so decay at both infinities gives $U=U_0e^{-|y|/L_R}$. A <velocity> jump would introduce a derivative of a delta function, absent from $\bar Q_y$, so $U$ is continuous. Integrating through zero now gives $-[U']=\Delta Q$, and therefore
$$
\boxed{U_0=\frac{L_R\Delta Q}{2},\qquad U(y)=\frac{L_R\Delta Q}{2}e^{-|y|/L_R}.}
$$
For the regular interfacial <normal mode> $\psi'=\phi(y)e^{ik(x-ct)}$, the linear equation reads
$$
(U-c)\bigl(\phi''-(k^2+L_R^{-2})\phi\bigr)+\Delta Q\,\delta(y)\phi=0.
$$
In each region, choose zero perturbation <potential vorticity>, giving $\phi''-(k^2+L_R^{-2})\phi=0$. This supplies a regular solution even if $U=c$ at isolated locations away from the interface. Continuity and decay require $\phi=Ae^{-\gamma|y|}$, where $\gamma=\sqrt{k^2+L_R^{-2}}$. Integrating the equation across zero gives
$$
(U_0-c)[\phi']+\Delta Q\phi(0)=0.
$$
Since $[\phi']=-2\gamma A$, the nonzero <normal mode> has
$$
\boxed{c=U_0-\frac{\Delta Q}{2\gamma}=U_0\left(1-\sqrt{\frac{L_R^{-2}}{k^2+L_R^{-2}}}\right),\qquad \phi=Ae^{-\gamma|y|}.}
$$
This <potential-vorticity step edge wave> is an interfacial <Rossby wave>. Displacing the <potential vorticity> jump creates alternating anomalies; their inverted <velocity> displaces the interface further with an intrinsic propagation opposite to the positive mean current. The mechanism requires a nonzero jump, or more generally a gradient, of the mean <potential vorticity>. It does not require planetary variation of $f$. Reversing the jump reverses the intrinsic propagation; setting the jump to zero removes this interfacial restoring mechanism.
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