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