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Set $c=\sqrt{g{H}_0}$ and seekThe two remaining prognostic equations implyThe $x$ momentum equation is now geostrophic:ThereforeBoundedness for $x>0$ selects $\gamma =\mathrm{sgn}(f)c$ andBecause $F$ depends on $y+\gamma t$, the wave travels with meridional phase speed $-\gamma$. Along this western boundary it travels southward in the Northern Hemisphere and northward in the Southern Hemisphere, keeping the coast on the dynamically required side. This is a coastal Kelvin wave.

Moreover,Since $u=0$, its linear potential-vorticity disturbance is

Solved by gpt-5.6-sol high.

Potential-vorticity conservation still determines the final surface throughThe unbounded problem only required decay or matching as $|y|$ and $x$ become large. The coast adds a boundary condition. In the adjusted geostrophic flow,so $\eta$ must be constant along the connected wall. Odd symmetry and the arbitrary common height datum set that constant to zero: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.