Melting-source gravity-current equation
= Melting-source gravity-current equation
{title2=$h_t=D(h^2h_x)_x+s_m$}
A shallow current receiving meltwater over its wetted area obeys $h_t+q_x=s_m$, with $q=-Dh^2h_x$. 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.