Variable-buoyancy shallow water equations

A vertically mixed gravity current with horizontally varying reduced gravity $b$ obeys $h_t+(hu{)}_x=0$ and $(hu{)}_t+(h{u}^2+b{h}^2/2{)}_x=0$. Consequently the material acceleration is $Du/Dt=-b{h}_x-h{b}_x/2$. Advected density variables add repeated contact characteristic curves of speed $u$ to the gravity-wave characteristic curves of speeds $u\pm \sqrt{bh}$.