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

Let $\mathbf{u}=(u,v)$ and let the layer thickness be $H=H_0+\eta$. The inviscid shallow water equations areThey follow from a homogeneous incompressible fluid with small aspect ratio, hydrostatic pressure, negligible vertical acceleration, horizontal velocity nearly uniform through the depth, a material free surface, and a rigid stationary bottom. Rotation may be represented by an f-plane or beta plane. The model is useful because many atmospheric and oceanic motions are horizontally much broader than their depth, while the free surface or an internal density interface still supports waves and potential vorticity dynamics.

Let $\zeta =v_x-u_y$. Taking the vertical curl of momentum giveswhere $Df/Dt=\beta v$ supplies the planetary-vorticity term when $f$ varies. Continuity gives $DH/Dt=-H\nabla \cdot \mathbf{u}$. Combining the two equations yields material conservation of shallow-water potential vorticity:

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