 Melting-source gravity-current equation

A shallow current receiving meltwater over its wetted area obeys $h_t+q_x=s_m$, with $q=-D{h}^2{h}_x$. The source is zero ahead of the advancing front. The Stefan condition determines normal ice-retreat speed $v_m$; the water-volume source is $s_m=({\rho }_i/{\rho }_w)v_m$, or simply $v_m$ in the equal-density approximation. Water volume and geometric excavation must be distinguished if densities differ.