Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 345 / 2 / b / Solution

Define the concentration-dependent reduced gravityUnder the Boussinesq approximation, the three shallow water equations can be written in advective form asThe coefficient matrix of this quasilinear system has characteristic speedsThus this is a hyperbolic system when $g^{\prime }h>0$. Along the intermediate characteristic curve $dx/dt=u$, the concentration obeysIt decreases because ambient entrainment dilutes the chemical; detrainment does not change the concentration of a well-mixed parcel.