Subglacial current with constant viscous wall layers ($D={\rho }_{w}g{\delta }_v/(2\mu )$)

In a water layer confined between horizontal rock and a movable constant-pressure ice roof, hydrostatic pressure and two wall stresses $\mu u/{\delta }_v$ give $u=-Dh{h}_x$, $q=-D{h}^2{h}_x$ and $h_t=D(h^2{h}_x{)}_x$, with $D={\rho }_{w}g{\delta }_v/(2\mu )>0$. This closure uses a constant viscous boundary layer thickness, not a generic quadratic turbulent drag law.

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.

For constant water-production rate per wetted area $s_m$, dry zero-flux noses and footprint length $\mathcal{L}(t)$, the moving-boundary conservation of mass gives $\dot{\mathcal{V}}=s_m\mathcal{L}$ per out-of-plane span. Consequently $\ddot{\mathcal{V}}=s_m\dot{\mathcal{L}}$. Equivalently the melt volume is $s_m{\int }_0^t\mathcal{L}(t\prime )dt\prime$; a point begins producing melt only after the front arrives.