 Friction-dominated grounding-line evolution law

For a bed depth $b=\alpha x$, a thin subglacial till layer of thickness $l$ is idealized by Newtonian till lubrication, with dynamic viscosity $\lambda \mu$ and basal drag $\lambda \mu u/l$. Neglecting longitudinal stress divergence in the bulk yields $u=-(l/\lambda )(g/\nu )H{h}_x$. Retaining the unbuttressed Newtonian grounding-line stress condition as boundary data and differentiating ice-sheet flotation along the moving grounding line givesIndeed, mass conservation gives $H_t=-u{H}_x-H{u}_x$, $H_x=h_x+\alpha$, and ice-sheet flotation gives $H_t=(\alpha {\rho }_w/\rho -H_x){\dot{x}}_G$. This is a formal local friction closure, not a uniformly valid removal of membrane stress near every grounding line. If the coefficient of ${\dot{x}}_G$ vanishes, it is an implicit compatibility condition and cannot be divided to obtain a finite speed.