 Entraining shallow-water current in a triangular channel

For a prismatic channel of width $b(z)=z$, a Boussinesq current of depth $h$ occupies area $A=h^2/2$. Ambient fluid at rest enters across the interface at speed $w_e$, supplying volume but no horizontal momentum or excess density. The volume conservation, momentum and excess-density balances areHere $g^{\prime }$ is the reduced gravity. With $D={\partial }_t+u{\partial }_x$, these give $Dh+(h/2)u_x=w_e$, $Du+g^{\prime }{h}_x+(h/3)g_x^{\prime }=-2u{w}_e/h$, and $D{g}^{\prime }=-2g^{\prime }{w}_e/h$. The momentum source in primitive form describes acceleration of newly entrained fluid, even when bed drag vanishes.