Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 345 / 2

Assume a hydrostatic, vertically well-mixed, inviscid shallow layer, negligible ambient horizontal velocity, and uniform channel width. Volume, mass, and horizontal momentum balances areCombining the first two givesDetrainment removes fluid with the layer's own density and velocity, so it cancels from the corresponding material density and velocity equations; entrainment dilutes and exerts a momentum load.

Define the reduced gravityand use the Boussinesq limit ${\rho }_1\simeq {\rho }_2$ except in buoyancy. The balances becomeThus, for $\mathbf{q}=(g^{\prime },h,u{)}^T$,which is the required entraining shallow-water layer system.

Put $c=\sqrt{g^{\prime }h}$. The three real eigenvalues areso the system is strictly hyperbolic for $g^{\prime }h>0$. Along $dx/dt=u$,Left eigenvectors for ${\lambda }_{\pm }$ are $(\pm h/(2c),\pm c/h,1)$, hencewhere $D_{\pm }={\partial }_t+(u\pm c){\partial }_x$.

When $w_e,w_d\to 0$, reduced gravity is materially conserved. If it is initially uniform, it remains constant and the other two relations integrate to the standard Riemann invariants