= Solution
The horizontal velocities $u$ and $v$ point east and north, $\eta$ is the displacement of the free surface from its mean level, $H_0$ is the undisturbed depth, $g$ is gravitational acceleration, and $f$ is the constant <Coriolis parameter> on an <f-plane>. The three <linearized shallow water equations> are horizontal momentum balance and <mass conservation>:
$$
u_t-fv=-g\eta_x,\qquad
v_t+fu=-g\eta_y,\qquad
\eta_t+H_0(u_x+v_y)=0.
$$
They follow from the rotating <Navier-Stokes equation> by assuming an inviscid homogeneous layer, <hydrostatic pressure>, horizontal scales much larger than $H_0$, depth-independent horizontal velocity, a flat impermeable bottom, constant $f$, and small surface displacement and velocity so that nonlinear products are neglected.
In a steady state, <geostrophic balance> gives
$$
\boxed{u_g=-\frac gf\eta_y},
\qquad
\boxed{v_g=\frac gf\eta_x}.
$$
The relative vorticity is $\zeta=v_x-u_y$. Expanding the <shallow-water potential vorticity> $(f+\zeta)/(H_0+\eta)$ to first order gives
$$
\frac f{H_0}+\frac1{H_0}
\left(\zeta-\frac f{H_0}\eta\right).
$$
Thus one convenient normalization of its disturbance is
$$
\boxed{q=\zeta-\frac f{H_0}\eta}.
$$
Taking the curl of momentum and using continuity shows $q_t=0$.
Solved by gpt-5.6-sol high.
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