Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 333 / 1

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: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 givesThe relative vorticity is $\zeta =v_x-u_y$. Expanding the shallow-water potential vorticity $(f+\zeta )/(H_0+\eta )$ to first order givesThus one convenient normalization of its disturbance isTaking the curl of momentum and using continuity shows $q_t=0$.

Solved by gpt-5.6-sol high.

Initially $\zeta =0$, so conservation of the linear potential vorticity givesThe final flow is in geostrophic balance, henceConsequently the adjusted height solves the modified Helmholtz equationwhere $L_R$ is the Rossby deformation radius.

The initial condition is independent of $x$ and odd in $y$. The bounded solution that is continuously differentiable at $y=0$ isIt givesDuring 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.

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.