= Solution
Potential-vorticity conservation still determines the final surface through
$$
\left(\nabla^2-L_R^{-2}\right)\eta_f
=-L_R^{-2}\eta_0\operatorname{sgn}(y),
\qquad x>0.
$$
The unbounded problem only required decay or matching as $|y|$ and $x$ become large. The coast adds a boundary condition. In the adjusted geostrophic flow,
$$
u(0,y)=-\frac gf\eta_y(0,y)=0,
$$
so $\eta$ must be constant along the connected wall. Odd symmetry and the arbitrary common height datum set that constant to zero:
$$
\boxed{\eta_f(0,y)=0}.
$$
This is a Dirichlet condition on $\eta$, even though it originated as no normal flow.
The initial disturbance does not instantaneously know this along the whole coast. A southward coastal <Kelvin wave> for $f>0$ carries the pressure signal and establishes the constant wall height behind its wavefront. In the final streamline sketch, the <quasi-geostrophic streamfunction> is proportional to $\eta_f$. Far from the wall the contours and current resemble the unbounded east--west front; near $x=0$ those contours bend through a right angle and run along the coast, so the incident geostrophic current turns into a southward boundary current instead of crossing the wall.
Solved by gpt-5.6-sol high.
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