Initially $\zeta=0$, so conservation of the linear <potential vorticity> gives
$$
\zeta_f-\frac f{H_0}\eta_f
=-\frac f{H_0}\eta_i.
$$
The final flow is in <geostrophic balance>, hence
$$
\zeta_f=\frac gf\nabla^2\eta_f.
$$
Consequently the adjusted height solves the modified Helmholtz equation
$$
\boxed{
\left(\nabla^2-\frac1{L_R^2}\right)\eta_f
=-\frac{\eta_i}{L_R^2}},
\qquad
\boxed{L_R=\frac{\sqrt{gH_0}}{|f|}},
$$
where $L_R$ is the <Rossby deformation radius>.

The initial condition is independent of $x$ and odd in $y$. The bounded solution that is continuously differentiable at $y=0$ is
$$
\boxed{
\eta_f(y)=\eta_0\operatorname{sgn}(y)
\left(1-e^{-|y|/L_R}\right)}.
$$
It gives
$$
\boxed{
u_f(y)=-\frac{g\eta_0}{fL_R}e^{-|y|/L_R}},
\qquad
\boxed{v_f=0}.
$$
During <geostrophic adjustment>, the part of the initial energy incompatible with the conserved potential-vorticity distribution radiates away as <inertia-gravity waves>. The remaining current has width $O(L_R)$ and is geostrophically balanced.

Solved by gpt-5.6-sol high.

Solution

Paper 333 / 1 / b / Solution

{wiki}

Geostrophic adjustment is the radiation of inertia-gravity waves from an unbalanced disturbance, leaving a slower flow in <geostrophic balance> while conserving linearized <potential vorticity>.

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 Incoming links: Geostrophic adjustment