Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 333 / 1 / ii

With no $y$ dependence, the linearized shallow water equations areTheir conserved linear potential-vorticity anomaly isbecause the initial velocity jump has derivative $2v_0\delta (x)$. The final steady state is in geostrophic balance, so $u=0$ and $fv=g{\eta }_x$. HenceTaking $f>0$ for definiteness and requiring decay at infinity gives

Initially, the potential energy is zero and, per unit length in $y$,For $L\gg R_d$, the final kinetic and potential energies are equal:Thus $E_{msteady}=\rho H_0{v}_0^2{R}_d$. During geostrophic adjustment, inertia-gravity waves carry the excess energy out of $|x|<L$; energy in that finite region is therefore not conserved. Their long-wave speed is $c=\sqrt{g{H}_0}$, so the adjustment of the stated region takes