Define the <reduced gravity>
$$
g'=g\frac{\rho_1-\rho_2}{\rho_2},
$$
and use the Boussinesq limit $\rho_1\simeq\rho_2$ except in buoyancy. The balances become
$$
g'_t+ug'_x=-g'\frac{w_e}{h},
$$
$$
h_t+uh_x+hu_x=w_e-w_d,
$$
$$
u_t+uu_x+g'h_x+\frac h2g'_x=-u\frac{w_e}{h}.
$$
Thus, for $\mathbf q=(g',h,u)^T$,
$$
\boxed{
\mathbf q_t+
\begin{pmatrix}
u&0&0\\
0&u&h\\
h/2&g'&u
\end{pmatrix}\mathbf q_x
=
\begin{pmatrix}
-g'w_e/h\\
w_e-w_d\\
-uw_e/h
\end{pmatrix}},
$$
which is the required <entraining shallow-water layer> system.

Solution

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

An entraining shallow-water layer changes depth, density, and momentum as ambient fluid crosses its interface. Entrainment dilutes its reduced gravity and accelerates or decelerates it through the momentum needed to bring new fluid into the layer.


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 Incoming links: Entraining shallow-water layer